precalculus homework 1 3

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Math Courses

Math 1060q — precalculus (fall 2024).

• Supplementary Materials

Course Coordinator (Storrs): Erika Fiore

Description:  Precalculus is a  preparation for calculus which includes a thorough review of algebra. Emphasis will be on functions and their applications; in particular, polynomials, rational functions, exponentials, logarithms, and trigonometric functions.

Prerequisites: A qualifying score of 17 on the mathematics placement exam (MPE), unless you began attending UConn prior to Fall 2016 (in which case it is still recommended). Students who fail to achieve this minimum score are required to spend time on the preparatory and learning modules before re-taking the MPE or register for a lower level Mathematics course. Not open for credit to students who have passed MATH 1120, 1125Q , or 1131Q . Students may not receive credit for this course and MATH 1040Q .

General Learning Objectives:

• Use definitions, formulas, properties, identities, theorems, and basic algebra skills to solve problems.
• Identify, graph, evaluate, describe, combine, compose, transform, invert, and analyze different types of functions (including polynomial, piecewise, rational, exponential, logarithmic, and trigonometric functions)
• Solve different types of equations (including polynomial, rational, exponential, logarithmic and trigonometric equations).
• Use functions to model and solve real world problems.

Course Materials:

Textbook: Precalculus 10e , by Larson (ISBN 978-1-337-27107-3)

You are  not  required to have a physical textbook. You are only required to have WebAssign access.

If you are participating in the  Husky Book Bundle Program, you do not need to purchase anything. Please sign into HuskyCT to access course materials through the WebAssign link.

If you have opted out of the Husky Book Bundle Program, you can buy the textbook from the UConn Bookstore, or direct from Cengage. You will need a WebAssign access code to access your homework assignments. When you buy the textbook from the bookstore, the WebAssign code will come bundled with the textbook. You will have two weeks of free access  to WebAssign, which includes the e-book, so you can get started right away in case you need some time to arrange to buy textbook with the access code.

There is only ONE way to access WebAssign .  Once the semester officially begins, simply log into your HuskyCT account ( lms.uconn.edu ), navigate to the page for Math 1060Q, then follow the link  for WebAssign Homework.  When logging into WebAssign (through HuskyCT),  do not use Internet Explorer or Safari. Use Firefox or Chrome.

If you need help getting started with WebAssign, please visit this link for more information and support from Cengage representatives:  https://www.cengage.com/coursepages/UConn_MATH1060Q

WebAssign assignments will be due every Tuesday and Thursday night. The first WebAssign assignment is due Tuesday, September 3rd , so please get started ASAP!

Calculators : The use of calculators IS  NOT permitted on exams or quizzes.

Classwork Assignments:

Quizzes and Exams:

Grading Scale:

• If you’ve taken precalculus before, be warned —  this course is harder . We will likely cover more material, and it will be more in-depth, than what you’ve done before.
• Don’t miss class! Each day builds on the previous days, so if you miss class, you get behind very quickly. If you do get sick or have to miss class, talk to your classmates and instructor to catch up before the next class. The outline for the course that is available using the link above will provide you with information about the topics to be covered in lecture.
• Do not skip the WebAssign homework. Doing these problems will give you the practice you need in order to be successful on quizzes and exams.
• Seek help early if you think you may need it! Some great resources for help are your instructor’s office hours, the  Q-Center , a tutor, and other students.

Academic Integrity and Honesty:

This course expects all students to act in accordance with the Guidelines for Academic Integrity at the University of Connecticut. UConn defines academic misconduct as “dishonest or unethical academic behavior that includes, but is not limited to, misrepresenting mastery in an academic area (e.g., cheating), failing to properly credit information, research, or ideas to their rightful originators or representing such information, research, or ideas as your own (e.g., plagiarism).” Below is a list of examples of academic misconduct in this course:

• Cheating or helping others cheat on a quiz or an exam. Using unauthorized materials on quizzes or exams including looking at your neighbors’ papers, your notes, books, or electronic devices is cheating. Communicating with any other human being (besides the instructor) during a quiz or an exam is cheating.
• Cheating or helping others cheat on homework.  It is okay to receive help from a classmate or tutor on homework, but all work submitted should represent your honest understanding of the material.
• Distributing your instructor’s materials to students’ outside of this course, including posting materials on the internet. You are forbidden from sharing or posting exams, quizzes, and homework problems created by your instructor. You are also forbidden from sharing or posting solutions to exams, quizzes, and homework, and other assignments.

Any student who commits an act of misconduct will be reported to the university. Violations of this policy may range from a zero on the assignment to a grade of F in the course. If you have questions about academic integrity or intellectual property, you should consult with your instructor.

For more details, click here: Policy on Academic, Scholarly, and Professional Integrity and Misconduct

Students with Disabilities:

The University of Connecticut is committed to protecting the rights of individuals with disabilities and assuring that the learning environment is accessible.  If you anticipate or experience physical or academic barriers based on disability or pregnancy, please let me know immediately so that we can discuss options. Students who require accommodations should contact the Center for Students with Disabilities, Wilbur Cross Building Room 204, (860) 486-2020, or  http://csd.uconn.edu/ .

To provide students with disabilities the appropriate accommodations, CSD must send an official letter to your instructor. If your accommodations require you to take quizzes and/or exams in a separate space, it is your responsibility to schedule an appointment with CSD’s testing center through MyAccess at least one week before each quiz/exam date.

Resources for Students Experiencing Distress:

The University of Connecticut is committed to supporting students in their mental health, their psychological and social well-being, and their connection to their academic experience and overall wellness. The university believes that academic, personal, and professional development can flourish only when each member of our community is assured equitable access to mental health services. The university aims to make access to mental health attainable while fostering a community reflecting equity and diversity and understands that good mental health may lead to personal and professional growth, greater self-awareness, increased social engagement, enhanced academic success, and campus and community involvement.

Students who feel they may benefit from speaking with a mental health professional can find support and resources through the Student Health and Wellness-Mental Health (SHaW-MH) office located in Storrs on the main campus in the Arjona Building, 4th Floor. Through SHaW- MH, students can make an appointment with a mental health professional and engage in confidential conversations or seek recommendations or referrals for any mental health or psychological concern. Mental health services are included as part of the university’s student health insurance plan and partially funded through university fees. If you do not have UConn’s student health insurance plan, most major insurance plans are also accepted. Please visit https://studenthealth.uconn.edu/ or call (860) 486-4705 if you have questions.

1.3 Section Exercises

1. Can the average rate of change of a function be constant?

Yes, the average rate of change of all linear functions is constant.

2. If a function[latex]\text{ }f\text{ }[/latex]is increasing on[latex]\text{ }\left(a,b\right)\text{ }[/latex]and decreasing on[latex]\text{ }\left(b,c\right),\text{ }[/latex]then what can be said about the local extremum of[latex]\text{ }f\text{ }[/latex]on[latex]\text{ }\left(a,c\right)?\text{ }[/latex]

3. How are the absolute maximum and minimum similar to and different from the local extrema?

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

4. How does the graph of the absolute value function compare to the graph of the quadratic function,[latex]\text{ }y={x}^{2},\text{ }[/latex]in terms of increasing and decreasing intervals?

For the following exercises, find the average rate of change of each function on the interval specified for real numbers [latex]\text{ }b\text{ }[/latex]or[latex]\text{ }h.[/latex]

5. [latex]f\left(x\right)=4{x}^{2}-7\text{ }[/latex]on[latex]\text{ }\left[1,\text{ }b\right][/latex]

[latex]4\left(b+1\right)[/latex]

6. [latex]g\left(x\right)=2{x}^{2}-9\text{ }[/latex]on[latex]\text{ }\left[4,\text{ }b\right][/latex]

7. [latex]p\left(x\right)=3x+4\text{ }[/latex]on[latex]\text{ }\left[2,\text{ }2+h\right][/latex]

8. [latex]k\left(x\right)=4x-2\text{ }[/latex]on[latex]\text{ }\left[3,\text{ }3+h\right][/latex]

9. [latex]f\left(x\right)=2{x}^{2}+1\text{ }[/latex]on[latex]\text{ }\left[x,x+h\right][/latex]

[latex]4x+2h[/latex]

10. [latex]g\left(x\right)=3{x}^{2}-2\text{ }[/latex]on[latex]\text{ }\left[x,x+h\right][/latex]

11. [latex]a\left(t\right)=\frac{1}{t+4}\text{ }[/latex]on[latex]\text{ }\left[9,9+h\right][/latex]

[latex]\frac{-1}{13\left(13+h\right)}[/latex]

12. [latex]b\left(x\right)=\frac{1}{x+3}\text{ }[/latex]on[latex]\text{ }\left[1,1+h\right][/latex]

13. [latex]j\left(x\right)=3{x}^{3}\text{ }[/latex]on[latex]\text{ }\left[1,1+h\right][/latex]

[latex]3{h}^{2}+9h+9[/latex]

14. [latex]r\left(t\right)=4{t}^{3}\text{ }[/latex]on[latex]\text{ }\left[2,2+h\right][/latex]

15. [latex]\frac{f\left(x+h\right)-f\left(x\right)}{h}\text{ }[/latex]given[latex]\text{ }f\left(x\right)=2{x}^{2}-3x\text{ }[/latex]on[latex]\text{ }\left[x,x+h\right][/latex]

[latex]4x+2h-3[/latex]

For the following exercises, consider the graph of[latex]\text{ }f\text{ }[/latex]shown in (Figure) .

16. Estimate the average rate of change from[latex]\text{ }x=1\text{ }[/latex]to[latex]\text{ }x=4.[/latex]

17. Estimate the average rate of change from[latex]\text{ }x=2\text{ }[/latex]to[latex]\text{ }x=5.[/latex]

[latex]\frac{4}{3}[/latex]

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

increasing on[latex]\text{ }\left(-\infty ,-2.5\right)\cup \left(1,\infty \right),\text{ }[/latex]decreasing on[latex]\text{ }\left(-2.5,\text{ }1\right)[/latex]

increasing on[latex]\text{ }\left(-\infty ,1\right)\cup \left(3,4\right),\text{ }[/latex]decreasing on[latex]\text{ }\left(1,3\right)\cup \left(4,\infty \right)[/latex]

For the following exercises, consider the graph shown in (Figure) .

22. Estimate the intervals where the function is increasing or decreasing.

23. Estimate the point(s) at which the graph of[latex]\text{ }f\text{ }[/latex]has a local maximum or a local minimum.

local maximum:[latex]\text{ }\left(-3,\text{ }50\right),\text{ }[/latex]local minimum:[latex]\text{ }\left(3,\text{ }-50\right)\text{ }[/latex]

For the following exercises, consider the graph in (Figure) .

24. If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.

25. If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

absolute maximum at approximately[latex]\text{ }\left(7,\text{ }150\right),\text{ }[/latex]absolute minimum at approximately[latex]\text{ }\left(-7.5,\text{ }-220\right)[/latex]

26. (Figure) gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?

27. (Figure) gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

a. –3000; b. –1250

For the following exercises, find the average rate of change of each function on the interval specified.

28. [latex]f\left(x\right)={x}^{2}\text{ }[/latex]on[latex]\text{ }\left[1,\text{ }5\right][/latex]

29. [latex]h\left(x\right)=5-2{x}^{2}\text{ }[/latex]on[latex]\text{ }\left[-2,\text{4}\right][/latex]

30. [latex]q\left(x\right)={x}^{3}\text{ }[/latex]on[latex]\text{ }\left[-4,\text{2}\right][/latex]

31. [latex]g\left(x\right)=3{x}^{3}-1\text{ }[/latex]on[latex]\text{ }\left[-3,\text{3}\right][/latex]

Show Solution

32. [latex]y=\frac{1}{x}\text{ }[/latex]on[latex]\text{ }\left[1,\text{ 3}\right][/latex]

33. [latex]p\left(t\right)=\frac{\left({t}^{2}-4\right)\left(t+1\right)}{{t}^{2}+3}\text{ }[/latex]on[latex]\text{ }\left[-3,\text{1}\right][/latex]

34. [latex]k\left(t\right)=6{t}^{2}+\frac{4}{{t}^{3}}\text{ }[/latex]on[latex]\text{ }\left[-1,3\right][/latex]

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

35. [latex]f\left(x\right)={x}^{4}-4{x}^{3}+5[/latex]

Local minimum at[latex]\text{ }\left(3,-22\right),\text{ }[/latex]decreasing on[latex]\text{ }\left(-\infty ,\text{ }3\right),\text{ }[/latex]increasing on[latex]\text{ }\left(3,\text{ }\infty \right)\text{ }[/latex]

36. [latex]h\left(x\right)={x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}-1[/latex]

37. [latex]g\left(t\right)=t\sqrt{t+3}[/latex]

Local minimum at[latex]\text{ }\left(-2,-2\right),\text{ }[/latex]decreasing on[latex]\text{ }\left(-3,-2\right),\text{ }[/latex]increasing on[latex]\text{ }\left(-2,\text{ }\infty \right)[/latex]

38. [latex]k\left(t\right)=3{t}^{\frac{2}{3}}-t[/latex]

39. [latex]m\left(x\right)={x}^{4}+2{x}^{3}-12{x}^{2}-10x+4[/latex]

Local maximum at[latex]\text{ }\left(-0.5,\text{ }6\right),\text{ }[/latex]local minima at[latex]\text{ }\left(-3.25,-47\right)\text{ }[/latex]and[latex]\text{ }\left(2.1,-32\right),\text{ }[/latex]decreasing on[latex]\text{ }\left(-\infty ,-3.25\right)\text{ }[/latex]and[latex]\text{ }\left(-0.5,\text{ }2.1\right),\text{ }[/latex]increasing on[latex]\text{ }\left(-3.25,\text{ }-0.5\right)\text{ }[/latex]and[latex]\text{ }\left(2.1,\text{ }\infty \right)\text{ }[/latex]

40. [latex]n\left(x\right)={x}^{4}-8{x}^{3}+18{x}^{2}-6x+2[/latex]

The graph of the function[latex]\text{ }f\text{ }[/latex]is shown in (Figure 18) .

41. Based on the calculator screen shot, the point[latex]\text{ }\left(1.333,\text{ }5.185\right)\text{ }[/latex] is which of the following?

• a relative (local) maximum of the function
• the vertex of the function
• the absolute maximum of the function
• a zero of the function

42. Let [latex]f\left(x\right)=\frac{1}{x}.[/latex] Find a number[latex]\text{ }c\text{ }[/latex]such that the average rate of change of the function[latex]\text{ }f\text{ }[/latex]on the interval[latex]\text{ }\left(1,c\right)\text{ }[/latex]is[latex]\text{ }-\frac{1}{4}.[/latex]

43. Let[latex]\text{ }f\left(x\right)=\frac{1}{x}[/latex]. Find the number[latex]\text{ }b\text{ }[/latex]such that the average rate of change of[latex]\text{ }f\text{ }[/latex]on the interval[latex]\text{ }\left(2,b\right)\text{ }[/latex]is[latex]\text{ }-\frac{1}{10}.[/latex]

[latex]b=5[/latex]

Real-World Applications

44. At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?

45. A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

2.7 gallons per minute

46. Near the surface of the moon, the distance that an object falls is a function of time. It is given by[latex]\text{ }d\left(t\right)=2.6667{t}^{2},\text{ }[/latex]where[latex]\text{ }t\text{ }[/latex]is in seconds and[latex]\text{ }d\left(t\right)\text{ }[/latex]is in feet. If an object is dropped from a certain height, find the average velocity of the object from[latex]\text{ }t=1\text{ }[/latex]to[latex]\text{ }t=2.[/latex]

47. The graph in (Figure) illustrates the decay of a radioactive substance over[latex]\text{ }t\text{ }[/latex]days.

Use the graph to estimate the average decay rate from[latex]\text{ }t=5\text{ }[/latex]to[latex]\text{ }t=15.[/latex]

approximately –0.6 milligrams per day

Privacy Policy

MTH 161: PreCalculus I (Online)

Samuel Dominic Chukwuemeka (SamDom For Peace) B.Eng., A.A.T, M.Ed., M.S

Course Navigation Links

About online classes: taking mth 161 with mr. c.

• MLM Registration and Resources
• Course Objectives
• Discussion Board (DB)
• Frequently Asked Questions (FAQs)
• Welcome to Our Class: Greetings and Welcome Video
• Course Syllabus
• Course Announcements

Pacing Guides

• Grades and Grades Calculators

Student Evaluation of Faculty

Projects (please do only one).

Piecewise Functions: Power Bill

Piecewise Functions: Water Bill

Piecewise Functions: Federal Tax

Matrices: Cryptology

Past Course Syllabi

Course Syllabus: MTH 161-V01 (Summer 2023)

Welcome to Our Class

It depends. Online classes are convenient. Most of them are asynchronous, so you work at your own time/pace. However, they demand a lot of time especially if you are struggling with the subject/course. I am not trying to scare you. I am just being honest with you. There are several factors that probably made you choose an online class: family demands, extended family demands, and work demands among others. I understand all these demands. Hence, the course is set up for you to acquire the knowledge while maintaining a family – work – school balance. Be it as it may, I am more concerned with your successfully passing the course. This includes: (I.) Acquiring the knowledge: of the topics in Math for School Teachers (eText; Notes; Videos; Learning Aids: Help Me Solve This , View an Example ; Multimedia Resources among others) (II.) Discussing the knowledge: (Discussion Board) (III.) Applying the knowledge: (Project-based Learning), and (IV.) Mastering the knowledge: (successfully completing the MyLab Math assignments).

Notable Notes about this Course

MyLab Math (MLM) Registration and Resources (eText, Solutions Manual and Multimedia Resources)

Access the eText/accessible eText

Access the Student's Solutions Manual

Access the Multimedia Resources

Week 1 (Monday, 05/20 – Saturday, 05/25)

• Relations and Functions Solved Examples on the Overview, Domain, and Range of Functions Solved Examples on the Evaluation of Functions Solved Examples on the Arithmetic Operations on Functions Solved Examples on the Graphs of Functions Solved Examples on the Properties of Functions Solved Examples on the Difference Quotience of Functions

Week 2 (Sunday, 05/26 – Saturday, 06/01)

• All resources from Week 1 and Solved Examples on the Symmetry of Functions Solved Examples on Linear Functions Word Problems on Linear Functions Piecewise Functions Solved Examples on Piecewise Functions Solved Examples on the Graphs of Piecewise Functions Algebra Transformations Solved Examples on the Transformations of Functions

Week 3 (Sunday, 06/02 – Saturday, 06/08)

• All resources from Week 1, Week 2, and Solved Examples on Linear Models Texas Instruments (TI) Calculators for Linear Models Solved Examples on Functions (all topics)

Week 4 (Sunday, 06/09 – Saturday, 06/15)

• All resources from Week 1 through Week 3, and Solved Examples on Quadratic Functions Word Problems on Quadratic Functions Solved Examples on Quadratic Models Solved Examples on Quadratic Inequalities

Week 5 (Sunday, 06/16 – Saturday, 06/22)

• All resources from Week 1 through Week 4, and Polynomials Solved Examples on Polynomial Graphs Texas Instruments (TI) Calculators for Polynomial Problems Add, Subtract, Multiply, and Divide Polynomials Evaluate Polynomials Factor Polynomials

Week 6 (Sunday, 06/23 – Saturday, 06/29)

• All resources from Week 1 through Week 5, and Rational Functions Solved Examples on Rational Functions

Week 7 (Sunday, 06/30 – Saturday, 07/06)

• All resources from Week 1 through Week 6, and Solved Examples on Polynomial Inequalities Solved Examples on Rational Inequalities Solved Examples on the Theorems of Polynomials Solved Examples on Polynomial Functions (All) Solved Examples on the Applications of Polynomial Functions

Week 8 (Sunday, 07/07 – Saturday, 07/13)

• All resources from Week 1 through Week 7, and Solved Examples on the Composition of Functions Solved Examples on the Inverses of Functions

Week 9 (Sunday, 07/14 – Saturday, 07/20)

• All resources from Week 1 through Week 8, and Exponents and Logarithms Laws of Exponents and Laws of Logarithms Exponential Functions and Logarithmic Functions

Week 10 (Sunday, 07/21 – Saturday, 07/27)

• All resources from Week 1 through Week 9, and Expand and Condense Logarithms Solved Examples on Exponential Expressions Solved Examples on Exponential Equations Solved Examples on Logarithmic Expressions Solved Examples on Logarithmic Equations Exponential and Logarithmic Functions Applications Matrix Algebra Solved Examples: Two-variable Linear Systems: Guass-Jordan Method Solved Examples: Three-variable Linear Systems: Guass-Jordan Method

Dear Students, Greetings to you. As you evaluate me and my teaching style on the Canvas course, I ask that you consider these questions in addition to the survey questions. Thank you for giving me the opportunity to teach the course. It was nice working with you. I wish you the best in your academic profession. Thank you! Samuel Dominic Chukwuemeka (SamDom For Peace) B.Eng., A.A.T, M.Ed., M.S Working together for success Grading Method: Grading Method Classroom/Learning Environment: Canvas course management system. Course Assessments: MyLab Math (MLM) assignments, Discussion Board (DB) assessments, and Project-based Learning assessment (Piecewise Function Project) Direct forms of communication: Student Engagement Hours/Live Sessions, Comments to your DB Posts and/or Responses, Individualized Zoom sessions, Emails, and Google Voice text messages and calls. Indirect forms of communication: Course announcements, Websites (Notes, Videos, and Resources among others). Course Contents Please review the VCCS Course Description and the VCCS Detailed Outline Those are the basic topics that VCCS/BRCC require that I teach. (1.) Did I cover those topics: teach the topics and/or provide resources for the topics? Resources include your textbooks (eText and resources), my websites (including notes/videos/resources) (2.) Did I cover other necessary topics that is relevant for you to succeed in your profession? (3.) Did the assessments demonstrate the application of the topics? (4.) Did the contents and assessments demonstrate important skills such as critical thinking, use of technology, creativity, and organization among others? Teaching and Learning (5.) Did you acquire any knowledge from me? (6.) Did you acquire sufficient knowledge, or more than sufficient knowledge from me? (7.) Did you acquire any knowledge from any of your colleagues because of how the course was set up? (8.) Did you acquire sufficient knowledge, or more than sufficient knowledge from any of your colleagues because of the way the course was set up? (9.) Did I provide effective feedback on the draft submissions and the main submissions of your project? Did you acquire any knowledge based on my feedback? (10.) Did I provide comments/effective feedback for your DB Post and/or Response? Did you acquire any knowledge based on the feedback? (11.) Did you get any feedback for any of the questions on your MLM assignments? Did you acquire any knowledge based on that feedback? (12.) Did any of the feedback help you improve in any way? In other words, was the feedback useful to you in any way? (13.) Did you have enough support to ensure the successful completion of the course? Were your questions answered? Did you have enough resources/learning aids? (14.) Did I provide a safe and conducive environment for learning? (15.) Was the Grading Method fair? Pacing Guide (16.) Were you given enough time to learn the contents? (17.) Were you given enough time to complete the assessments? Consider the fact that you were given two due dates for the MLM assignments and the Project; and that there was no penalty for late work after the initial due date. Professionalism (18.) In all our communication (both direct and indirect), did I act professionally? (19.) In all our communication (both direct and indirect), did I use a respectful tone? (20.) What do you like or dislike about our communication? Personality (21.) Based on your experience with me (taking the course with me and communicating with me among others), how would you describe my person? Do I give a lot of work? Do I give a lot of explanations? Do I have a lot of expectations for my students? Do I really want you to learn? Do I really want you to succeed? Do I ask a lot of questions? Do I answer questions with questions sometimes? During those times, please note that it is a teaching technique. It is never meant to disrespect you. I do not disrespect my students. I respect them. It is a technique to guide you to review directions/concepts, and explain what you do not understand. Would you take another course with me? Why or why not? Did your views/perception about me affect you in completing this course successfully or unsuccessfully?

Precalculus Help and Problems

Topics in precalculus will serve as a transition between algebra and calculus , containing material covered in advanced algebra and trigonometry courses. Precalculus consists of insights needed to understand calculus.

Still need help after using our precalculus resources? Use our service to find a precalculus tutor .

A brief overview of sets, one of the fundamental principles of mathematics. The topic of sets introduces grouping of objects and number classifications.

Exponential Functions

An extension of exponents in terms of functions, as well as introducing the constant e . Topics include exponential growth and decay, solving with logs, compound and continuously compounded interest, and the exponential function of e .

Logarithmic Functions

A more in depth look at logarithms and logarithmic functions, as well as how they relate to exponents. Topics include the standard and natural log, solving for x and the inverse properities of logarithms.

Radical Functions

An introduction to functions with square roots and radicals and how they relate to conic sections. Topics include functions finding zeros of radical functions, functions with square roots and higher roots, and functions with no solution.

Series and Sequences

Sequences and Series always go hand in hand and they introduce the concept of Mathematical Patterns and how to deal with them. This section deals with the different types of Series and Sequences as well as the methods of finding the next term in a sequence or the sum of a series.

The major types of series and sequences include:

Arithmetic Progression

Geometric Progression

Factorials, Permutations and Combinations

Introduction to the factorial notation. The concept of combinations and permutations is introduced and explained.

Binomial Theorem

Statement of the Binomial Theorem. Pascals’ Triangle and its relation to Binomial Theorem. Expanding polynomials using the Binomial Theorem.

Parametric Equations

An overview of the use of parametric equations, including parametrizing functions and finding a function for a set of parametric equations. Topics include parametrizing lines, segments, circles and ellipses, and peicewise functions.

Polar Coordinates

An introduction to the polar coordinate system. Topics include graphing points, converting from rectangular to polar and polar to rectangular coordinates, converting degrees and radians, and polar equations.

Introduction to the concept of matrices. Different types of matrices discussed. Matrix algebra including addition, subtraction and multiplication. Matrix inverses and determinants.

Subpages include:

Matrix Equality

Matrix Addition and Subtraction

Matrix Multiplication

Special Matrices

Inverse Matrix

Reduction to Row Echelon Form

Systems of Equations

Many times we come cross systems of equations and we’re usually at odds with how to solve them. Most times we deal with two-variable or three-variable systems of equations but the methods explained in this section can be used to solve any number variable system of equations. The trick is picking which one would work better and faster for you.

The methods explained include:

Substitution Method

Elimination Method

Matrix Method

Consistent and Inconsistent Systems

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Precalculus

Mrs. Snow's Math

Notes are intended to compliment the current text in use at McNeil High School Precalculus, Enhanced with Graphing Utilities, Texas Edition By Michael Sullivan and Michael Sullivan, III

FALL SEMESTER

Fundamentals

MATHXL FOR SCHOOL – where to find it on line and where are all the assignments!

Algebra II Review Annotated Notes

Chapter 2:  Functions and Their Graphs All assignments are due at the start of the next class period.  The first assignments of this school year will be completed on paper: 

Current assignments are currently not available on Mathxl, the precalculus online homework program.  For full credit, use separate paper:   please write your full name, class period, and identify the assignment with chapter.section number.  Write out each problem, show all work and circle the answer.

Lesson 2.1 and Lesson 2.2 Functions and 2.2 The Graph of a Function Mathxl Homework 2.1 and 2.2 Solutions 2.1 2.2 Mathxl -Textbook Homework 2.3 and 2.4 Solutions Textbook 2.3 and 2.4 Mathxl – Textbook Homework 2.5 Solutions 2.5 Annotated Notes Video Lesson 2.1 and 2.2 Lesson 2.3 and Lesson 2.4 Properties of Functions,   2.4  Piecewise and Greatest Integer Functions Annotated Notes 2.3 and 2.4 Video Lesson 2.3 and 2.4 Lesson 2.5   Graphing Techniques:  Transformations Annotated Notes 2.5 Video Lesson 2.5 In-class Transformation Practice Annotated Transformation Practice mrssnowsmath.com/…/chapter2review.2023r.pdf 2.5 Homework Worksheet Chapter 2 Review Review Solutions  

Chapter 4:  Polynomial and Rational Functions

Lesson 4.1  Polynomial Functions and Model Annotated Notes Video Lesson 4.1 Lesson 4.2 The Real Zeros of a Polynomial Function Lesson 4.3 Complex Zeros; Fundamental Theorem of Algebra Annotated Notes 4.2 and 4.3 Video Lesson 4.2 and 4.3 Lesson 4.4 and Lesson 4.5  4.4 Properties of Rational Functions and.4.5 The Graphs of a Rational Function Annotated Notes 4.4 and 4.5 Video Lesson 4.4 and 4.5 Lesson 4.6 Polynomial and Rational Inequalities Annotated Notes 4.6 Video Lesson 4.6 Chapter 4 Spiral Ch 2 Review    Updated 9/2023 Spiral Review Solutions    

Chapter 6 Trigonometric Functions

Quizlet Links: Sine, Cosine, and Tangent Radian Values Practice with radians from the unit circle; flashcards and more Unit Circle practice page   2 unit circles for drill practice

Lesson 6.1 and 6.2 Part 1 Angles and Their Measure, The Unit Circle Annotated Notes 6.1 And 6.2 Part 1 Video Lesson6.1 and 6.2 part 1 Lesson 6.1 and 6.2 Part 2 Trigonometric Functions: Unit Circle Approach Annotated Notes 6.1 and 6-2 Part2 Video Lesson 6.1 and 6.2 part 2 2 Unit Circles   Practice Form Lesson 6.3 Properties of the Trigonometric Functions Annotated Notes Video Lesson 6.3 Review 6.1-6.3 Chapter 6.1-6.3 Review Solutions   

Unit Circle Chart – In Order Lesson 6.4 Graphs of the Sine and Cosine Functions Annotated Notes 6.4 Video Lesson 6.4 Worksheet6.4 Lesson 6.5   Graphs of Secant and Cosecant Functions Annotated Notes 6.5 Part 1 Lesson 6.5 Graphs of Tangent and Cotangent Functions Annotated Notes 6.5 Part 2 Worksheet 6.5    Worksheet for: secant, Cosecant, Tangent and Cotangent Functions Video Lesson 6.5    this covers both part 1 and part 2:  Secant, Cosecant, Tangent and Cotangent Functions Lesson 6.6 Phase Shift Annotated Notes 6.6 Video Lesson 6.6 Lesson 8.5   Simple Harmonic Motion Annotated Notes 8.5 Video Lesson 8.5   “The Ferris Wheel Problem”   Video showing Simple Harmonic Motion Spiral Review 6.1-6.6, 8.5   Spiral Review Solutions

Lesson 5.1 and Lesson 5.2 Composite Functions and One-to-One Functions, Inverse Functions Annotated Notes 5.1 and 5.2 Video Lesson 5.1 and 5.2

Lesson 7.1 and 7.2   The Inverse Sine, Cosine, and Tangent Functions and The Inverse Trigonometric Function, Continued 7.1 and 7.2 Annotated Notes Video Lesson 7.1 Extra Examples Lesson 7.3 Trigonometric Equations Annotated Notes 7.3 Video Lesson 7.3 Review 5.1-5.2  and 7.1-7.3 Review 7.1-7.3 Solutions (revised 11/2/2019)

Unit Circle Sine, Cosine and Tangent values – Quizlet     link to a quizlet that has flash cards for our unit circle values.  The “Learn” link will quiz you with the flash cards out of order.

Trigonometric Identities Reference Sheet The first page will need to be memorized! Lesson 7.4 Trigonometric Identities Annotated Notes 7 youtube.com/watch .4 Lesson 7.4 Worksheet Proofs Trig Identities   fall 2022 we are only doing the odd problems.  The even are available for additional practice Lesson 7.5 Sum and Difference Formulas Annotated Notes 7.5 Video Lesson 7.5 Lesson 7.6 Double-angle and Half-angle Formulas Annotated Notes 7.6 Video Lesson 7.6 Review Spiral 5.1-2, 7.1-3 7.4-7.5 Review Answers    Corrected!

 Chapter 10

Lesson 10.1 and 10.2 Conics and the Parabola Annotated Notes 10.1 and 10.2    (also has a brief discussion of circles) Video Lesson 10.1 and 10.2

Lesson 10.3 The Ellipse Annota te d Notes 10.3 Lesson 10.4 The Hyperbola Annotated Notes 10.4 Lesson 10.7 Plane Curves and Parametric Equations Annotated Notes 10.7

Review Chapter 10    10.4, hyperbola is not included in this review, and only covers the parabola and ellipse centered about the origin. Review Solutions

Fall Final Exam Review Fall Final Exam solutions

GUIDELINE FOR NOTECARD

This link will take you to an excel spreadsheet that will allow you to take your averages for either fall or spring semester and see what you need for the grading period or final to pass class.  This is designed for the grading cycles at McNeil High School. “WHAT IF…”GRADE CHECK  for Fall and Spring Semesters

SPRING SEMESTER 

Chapter 8 Lesson 8.1 Right Triangle Trigonometry; Applications Annotated Notes 8.1 Video 8.1 Trigonometry Applications Worksheet Lesson 8.2 The Law of Sines Lesson 8.2 The Ambiguous Case Example Annotated Notes 8.2 Video 8.2 Lesson 8.3 The Law of Cosines Lesson 8.4 Area of a Triangle Annotated Notes 8.3 and 8.4 Video 8.3 and 8.4 Review Chapter 8 Review Chapter  Solutions  

Lesson 8.5 Simple Harmonic Motion (presented in the fall with Chapter 6) Annotated Notes 8.5 are located with Chapter 6

Chapter 9 Lesson 9.1 Polar Coordinates Annotated Notes 9.1 Video Lesson 9.1 Lesson 9.2 Polar Equations and Graphs Annotated Notes .9.2 Video Lesson 9.2 Limacon Examples Polar Graph Paper Lesson 9.4 Vectors Video Lesson 9.4 Annotated Notes 9.4 9.4 Static Equilibrium Problem #19 example Lesson 9.5 The Dot Product Annotated Notes 9.5 Video Lesson 9.5 Spiral Ch 8 and 9 Re view Spiral Review Solutions

Chapter 5 and 11.5 Lesson5.3 Exponential Functions Annotated Notes 5.3  and corrected Video Lesson 5.3 Lesson 5.4 Logarithmic Functions  and Lesson 5.5 Properties of Logarithms Annotated Notes 5.4 and 5.5 V ideo Lesson 5.4 Video Lesson 5.5

Lesson 5.6 Logarithmic and Exponential Equations Annotated Notes 5.6 Video Lesson 5.6 Lesson 5.7 and Lesson 5.8 Financial Models   and  Exponential Growth and Decay Models Annotated Notes 5.7 and 5.8 Video Lesson 5.7  Video Lesson 5.8 Chapter 5 Review  Review Solutions Chapter 5

Extra Lessons 11.5 and Rational Expressions Addition and Subtraction of Rational Expressions Annotated Notes Video Lesson Addition and Subtraction of Rational Expressions Multiplication and Division of Rational Expressions  Annotated Notes Video Lesson Multiplication and Division of Rational Expressions

Lesson 11.5 Partial Fraction Decomposition Annotated Notes 11.5 Video Lesson 11.5 

Chapter  12 Lesson 12.1 Sequences Annotated Notes 12.1 Video Lesson Lesson 12.2 Arithmetic Sequences Annotated Notes 12.2 Video Lesson Lesson 12.3   Geometric Sequences Lesson 12.3 Lecture Notes modified for voice over lecture Annotated Notes 12.3 Video Lesson Lesson 12.4 Mathematical Induction Lesson 12.4 Lecture Notes modified for voice over lecture Annotated Notes 12.4 12.4 Homework Worksheet Lesson 12.5  The Binomial Theorem Lesson 12.5 Lecture Notes modified for voice over lecture Annotated Notes 12.5 Video Lesson Review Chapter 12    

Solutions Review Ch 12 and 11.5   if you want credit for the review, make sure you show your work,  copying these answers will result in a 0%

Chapter 14 –  A Preview of Calculus  

Lesson 14.1 Finding Limits Using Tables and Graphs Annotated Notes 14.1 *    Video Lecture 14.1 14.1 Worksheet

Lesson 14.2 Algebra Techniques for Finding Limits *    Video Lecture 14.2 Annotated Notes 14.2 14.2 Worksheet

Lesson 14.3  The Tangent Problem:  The Derivative Annotated Notes 14.3 *     Video Lecture 14.3 14.3 Worksheet Helpful videos for visualizing secant lines morphing into a tangent line: The Tangent Line and the Derivative   an 11 minute video, all is very good, the definition by derivative starts at 4 minutes. Applet:  Ordinary Derivative by Limit Definition   as an interactive applet to let the student see as the distance between points goes to zero,  the secant line becomes the tangent line.

Lesson 14.4  Limits at Infinity Annotated Notes 14.4 14.4 Worksheet *    Video Lecture 14.4

Lesson 14.5 The Area Problem:  The Integral Annotated Notes 14.5 14.5 Worksheet *    Video Lecture 14.5

Solutions to Chapter 14 practice problems

14.3, problem #7: correct answer is f'(x)=-5

Spiral Review Chapter 14 and Chapter 12 -remember, no work no credit!! Review Ch 12 and 14 Solutions

Chapter 10 notes locate here are abridged lessons for Spring 2024 notes showing the center or vertices not at the origin may be found in the fall notes above

Chapter 10 Spring 2024 Lesson 10.2     The Parabola Annotated Notes 10.2 Lesson 10.3  The Ellipse Annotated Notes 10.3 Lesson 10.4   The Hyperbola Annotated Notes 10.4 Lesson 10.7  Plane Curves and Parametric Equations Annotated Notes 10.7 Chapter Review Chapter Review Solutions

Note Card For Precalculus Final All students, whether exempting the final exam or not,  are required to complete the Final Exam Review.  Work must be shown for credit.   Spring Final Exam Review    spring 2024 Spring Final Review Solutions    spring 2024      

Other Notes

Intro to Calculus Lesson on Derivatives and Integration Annotated Notes Derivatives Worksheet #1 – Product and Quotient Rules Worksheet #2 – Chain Rule Chain Rule Notes Worksheet #3 – Integrals Integral Notes

This link will take you to an excel spreadsheet that will allow you to take your averages for either fall or spring semester and see what you need for the grading period or final to pass class.  This is designed for the grading cycles at McNeil High School. “WHAT IF…”GRADE CHECK for 2020-2021 School Year

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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

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• The calculus section will carry out differentiation as well as definite and indefinite integration.
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1.1 Functions and Function Notation

Learning objectives.

In this section, you will:

• Determine whether a relation represents a function.
• Find the value of a function.
• Determine whether a function is one-to-one.
• Use the vertical line test to identify functions.
• Graph the functions listed in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range . Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

The domain is { 1 , 2 , 3 , 4 , 5 } . { 1 , 2 , 3 , 4 , 5 } . The range is { 2 , 4 , 6 , 8 , 10 } . { 2 , 4 , 6 , 8 , 10 } .

Note that each value in the domain is also known as an input value, or independent variable , and is often labeled with the lowercase letter x . x . Each value in the range is also known as an output value, or dependent variable , and is often labeled lowercase letter y . y .

A function f f is a relation that assigns a single value in the range to each value in the domain . In other words, no x -values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, { 1 , 2 , 3 , 4 , 5 } , { 1 , 2 , 3 , 4 , 5 } , is paired with exactly one element in the range, { 2 , 4 , 6 , 8 , 10 } . { 2 , 4 , 6 , 8 , 10 } .

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain, { even, odd } { even, odd } is not paired with exactly one element in the range, { 1 , 2 , 3 , 4 , 5 } . { 1 , 2 , 3 , 4 , 5 } . For example, the term “odd” corresponds to three values from the range, { 1 , 3 , 5 } { 1 , 3 , 5 } and the term “even” corresponds to two values from the range, { 2 , 4 } . { 2 , 4 } . This violates the definition of a function, so this relation is not a function.

Figure 1 compares relations that are functions and not functions.

A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input values make up the domain , and the output values make up the range .

Given a relationship between two quantities, determine whether the relationship is a function.

• Identify the input values.
• Identify the output values.
• If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

Determining If Menu Price Lists Are Functions

The coffee shop menu, shown in Figure 2 consists of items and their prices.

• ⓐ Is price a function of the item?
• ⓑ Is the item a function of the price?

Each item on the menu has only one price, so the price is a function of the item.

Therefore, the item is a not a function of price.

Determining If Class Grade Rules Are Functions

In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? Table 1 shows a possible rule for assigning grade points.

For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.

In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

Table 2 1 lists the five greatest baseball players of all time in order of rank.

• ⓐ Is the rank a function of the player name?
• ⓑ Is the player name a function of the rank?

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables h h for height and a a for age. The letters f , g , f , g , and h h are often used to represent functions just as we use x , y , x , y , and z z to represent numbers and A , B , A , B , and C C to represent sets.

Remember, we can use any letter to name the function; the notation h ( a ) h ( a ) shows us that h h depends on a . a . The value a a must be put into the function h h to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example f ( a + b ) f ( a + b ) means “first add a and b , and the result is the input for the function f .” The operations must be performed in this order to obtain the correct result.

Function Notation

The notation y = f ( x ) y = f ( x ) defines a function named f . f . This is read as “ y “ y is a function of x . ” x . ” The letter x x represents the input value, or independent variable. The letter y , y , or f ( x ) , f ( x ) , represents the output value, or dependent variable.

Using Function Notation for Days in a Month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.

The number of days in a month is a function of the name of the month, so if we name the function f , f , we write days = f ( month ) days = f ( month ) or d = f ( m ) . d = f ( m ) . The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

For example, f ( March ) = 31 , f ( March ) = 31 , because March has 31 days. The notation d = f ( m ) d = f ( m ) reminds us that the number of days, d d (the output), is dependent on the name of the month, m m (the input).

Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.

Interpreting Function Notation

A function N = f ( y ) N = f ( y ) gives the number of police officers, N , N , in a town in year y . y . What does f ( 2005 ) = 300 f ( 2005 ) = 300 represent?

When we read f ( 2005 ) = 300 , f ( 2005 ) = 300 , we see that the input year is 2005. The value for the output, the number of police officers ( N ) , ( N ) , is 300. Remember, N = f ( y ) . N = f ( y ) . The statement f ( 2005 ) = 300 f ( 2005 ) = 300 tells us that in the year 2005 there were 300 police officers in the town.

Use function notation to express the weight of a pig in pounds as a function of its age in days d . d .

Instead of a notation such as y = f ( x ), y = f ( x ), could we use the same symbol for the output as for the function, such as y = y ( x ), y = y ( x ), meaning “ y is a function of x ?”

Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as f , f , which is a rule or procedure, and the output y y we get by applying f f to a particular input x . x . This is why we usually use notation such as y = f ( x ) , P = W ( d ) , y = f ( x ) , P = W ( d ) , and so on.

Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.

Table 3 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function f f where D = f ( m ) D = f ( m ) identifies months by an integer rather than by name.

Table 4 defines a function Q = g ( n ) . Q = g ( n ) . Remember, this notation tells us that g g is the name of the function that takes the input n n and gives the output Q . Q .

Table 5 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

Given a table of input and output values, determine whether the table represents a function.

• Identify the input and output values.
• Check to see if each input value is paired with only one output value. If so, the table represents a function.

Identifying Tables that Represent Functions

Which table, Table 6 , Table 7 , or Table 8 , represents a function (if any)?

Table 6 and Table 7 define functions. In both, each input value corresponds to exactly one output value. Table 8 does not define a function because the input value of 5 corresponds to two different output values.

When a table represents a function, corresponding input and output values can also be specified using function notation.

The function represented by Table 6 can be represented by writing

Similarly, the statements

represent the function in Table 7 .

Table 8 cannot be expressed in a similar way because it does not represent a function.

Does Table 9 represent a function?

Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function f ( x ) = 5 − 3 x 2 f ( x ) = 5 − 3 x 2 can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

Given the formula for a function, evaluate.

• Substitute the input variable in the formula with the value provided.
• Calculate the result.

Evaluating Functions at Specific Values

Evaluate f ( x ) = x 2 + 3 x − 4 f ( x ) = x 2 + 3 x − 4 at:

• ⓒ a + h a + h
• ⓓ Now evaluate f ( a + h ) − f ( a ) h f ( a + h ) − f ( a ) h

Replace the x x in the function with each specified value.

• ⓐ Because the input value is a number, 2, we can use simple algebra to simplify. f ( 2 ) = 2 2 + 3 ( 2 ) − 4 = 4 + 6 − 4 = 6 f ( 2 ) = 2 2 + 3 ( 2 ) − 4 = 4 + 6 − 4 = 6
• ⓑ In this case, the input value is a letter so we cannot simplify the answer any further. f ( a ) = a 2 + 3 a − 4 f ( a ) = a 2 + 3 a − 4
• ⓒ With an input value of a + h , a + h , we must use the distributive property. f ( a + h ) = ( a + h ) 2 + 3 ( a + h ) − 4 = a 2 + 2 a h + h 2 + 3 a + 3 h − 4 f ( a + h ) = ( a + h ) 2 + 3 ( a + h ) − 4 = a 2 + 2 a h + h 2 + 3 a + 3 h − 4

and we know that

Now we combine the results and simplify.

Evaluating Functions

Given the function h ( p ) = p 2 + 2 p , h ( p ) = p 2 + 2 p , evaluate h ( 4 ) . h ( 4 ) .

To evaluate h ( 4 ) , h ( 4 ) , we substitute the value 4 for the input variable p p in the given function.

Therefore, for an input of 4, we have an output of 24.

Given the function g ( m ) = m − 4 , g ( m ) = m − 4 , evaluate g ( 5 ) . g ( 5 ) .

Solving Functions

Given the function h ( p ) = p 2 + 2 p , h ( p ) = p 2 + 2 p , solve for h ( p ) = 3. h ( p ) = 3.

If ( p + 3 ) ( p − 1 ) = 0 , ( p + 3 ) ( p − 1 ) = 0 , either ( p + 3 ) = 0 ( p + 3 ) = 0 or ( p − 1 ) = 0 ( p − 1 ) = 0 (or both of them equal 0). We will set each factor equal to 0 and solve for p p in each case.

This gives us two solutions. The output h ( p ) = 3 h ( p ) = 3 when the input is either p = 1 p = 1 or p = − 3. p = − 3. We can also verify by graphing as in Figure 6 . The graph verifies that h ( 1 ) = h ( − 3 ) = 3 h ( 1 ) = h ( − 3 ) = 3 and h ( 4 ) = 24. h ( 4 ) = 24.

Given the function g ( m ) = m − 4 , g ( m ) = m − 4 , solve g ( m ) = 2. g ( m ) = 2.

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation 2 n + 6 p = 12 2 n + 6 p = 12 expresses a functional relationship between n n and p . p . We can rewrite it to decide if p p is a function of n . n .

Given a function in equation form, write its algebraic formula.

• Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
• Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Finding the Algebraic Form of a Function

Express the relationship 2 n + 6 p = 12 2 n + 6 p = 12 as a function p = f ( n ) , p = f ( n ) , if possible.

To express the relationship in this form, we need to be able to write the relationship where p p is a function of n , n , which means writing it as p = [ expression involving n ] . p = [ expression involving n ] .

Therefore, p p as a function of n n is written as

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Expressing the Equation of a Circle as a Function

Does the equation x 2 + y 2 = 1 x 2 + y 2 = 1 represent a function with x x as input and y y as output? If so, express the relationship as a function y = f ( x ) . y = f ( x ) .

First we subtract x 2 x 2 from both sides.

We now try to solve for y y in this equation.

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function y = f ( x ) . y = f ( x ) .

If x − 8 y 3 = 0 , x − 8 y 3 = 0 , express y y as a function of x . x .

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation x = y + 2 y , x = y + 2 y , if we want to express y y as a function of x , x , there is no simple algebraic formula involving only x x that equals y . y . However, each x x does determine a unique value for y , y , and there are mathematical procedures by which y y can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for y y as a function of x , x , even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See Table 10 . 2

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function P . P . The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function P P at the input value of “goldfish.” We would write P ( goldfish ) = 2160. P ( goldfish ) = 2160. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function P P seems ideally suited to this function, more so than writing it in paragraph or function form.

Given a function represented by a table, identify specific output and input values.

• Find the given input in the row (or column) of input values.
• Identify the corresponding output value paired with that input value.
• Find the given output values in the row (or column) of output values, noting every time that output value appears.
• Identify the input value(s) corresponding to the given output value.

Evaluating and Solving a Tabular Function

Using Table 11 ,

• ⓐ Evaluate g ( 3 ) . g ( 3 ) .
• ⓑ Solve g ( n ) = 6. g ( n ) = 6.

• ⓐ Evaluating g ( 3 ) g ( 3 ) means determining the output value of the function g g for the input value of n = 3. n = 3. The table output value corresponding to n = 3 n = 3 is 7, so g ( 3 ) = 7. g ( 3 ) = 7.
• ⓑ Solving g ( n ) = 6 g ( n ) = 6 means identifying the input values, n , n , that produce an output value of 6. The table below shows two solutions: 2 2 and 4. 4.

When we input 2 into the function g , g , our output is 6. When we input 4 into the function g , g , our output is also 6.

Using the table from Evaluating and Solving a Tabular Function above, evaluate g ( 1 ) . g ( 1 ) .

Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Reading Function Values from a Graph

Given the graph in Figure 7 ,

• ⓐ Evaluate f ( 2 ) . f ( 2 ) .
• ⓑ Solve f ( x ) = 4. f ( x ) = 4.

Using Figure 7 , solve f ( x ) = 1. f ( x ) = 1.

Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in Table 12 .

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in Figure 1 (a) and Figure 1 (b) . The function in part (a) shows a relationship that is not a one-to-one function because inputs q q and r r both give output n . n . The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

One-to-One Function

A one-to-one function is a function in which each output value corresponds to exactly one input value.

Determining Whether a Relationship Is a One-to-One Function

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

A circle of radius r r has a unique area measure given by A = π r 2 , A = π r 2 , so for any input, r , r , there is only one output, A . A . The area is a function of radius r . r .

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure A A is given by the formula A = π r 2 . A = π r 2 . Because areas and radii are positive numbers, there is exactly one solution: A π . A π . So the area of a circle is a one-to-one function of the circle’s radius.

• ⓐ Is a balance a function of the bank account number?
• ⓑ Is a bank account number a function of the balance?
• ⓒ Is a balance a one-to-one function of the bank account number?

Evaluate the following: ⓐ If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade? ⓑ If so, is the function one-to-one?

Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value x x and the output value y , y , and we say y y is a function of x , x , or y = f ( x ) y = f ( x ) when the function is named f . f . The graph of the function is the set of all points ( x , y ) ( x , y ) in the plane that satisfies the equation y = f ( x ) . y = f ( x ) . If the function is defined for only a few input values, then the graph of the function is only a few points, where the x -coordinate of each point is an input value and the y -coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 10 tell us that f ( 0 ) = 2 f ( 0 ) = 2 and f ( 6 ) = 1. f ( 6 ) = 1. However, the set of all points ( x , y ) ( x , y ) satisfying y = f ( x ) y = f ( x ) is a curve. The curve shown includes ( 0 , 2 ) ( 0 , 2 ) and ( 6 , 1 ) ( 6 , 1 ) because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See Figure 11 .

Given a graph, use the vertical line test to determine if the graph represents a function.

• Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
• If there is any such line, determine that the graph does not represent a function.

Applying the Vertical Line Test

Which of the graphs in Figure 12 represent(s) a function y = f ( x ) ? y = f ( x ) ?

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 12 . From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x -values, a vertical line would intersect the graph at more than one point, as shown in Figure 13 .

Does the graph in Figure 14 represent a function?

Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test . Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

• Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
• If there is any such line, determine that the function is not one-to-one.

Applying the Horizontal Line Test

Consider the functions shown in Figure 12 (a) and Figure 12 (b) . Are either of the functions one-to-one?

The function in Figure 12 (a) is not one-to-one. The horizontal line shown in Figure 15 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in Figure 12 (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Is the graph shown in Figure 12 (c) one-to-one?

Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use x x as the input variable and y = f ( x ) y = f ( x ) as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in Table 13 .

Access the following online resources for additional instruction and practice with functions.

• Determine if a Relation is a Function
• Vertical Line Test
• Introduction to Functions
• Vertical Line Test on Graph
• One-to-one Functions
• Graphs as One-to-one Functions

1.1 Section Exercises

What is the difference between a relation and a function?

What is the difference between the input and the output of a function?

Why does the vertical line test tell us whether the graph of a relation represents a function?

How can you determine if a relation is a one-to-one function?

Why does the horizontal line test tell us whether the graph of a function is one-to-one?

For the following exercises, determine whether the relation represents a function.

{ ( a , b ) , ( c , d ) , ( a , c ) } { ( a , b ) , ( c , d ) , ( a , c ) }

{ ( a , b ) , ( b , c ) , ( c , c ) } { ( a , b ) , ( b , c ) , ( c , c ) }

For the following exercises, determine whether the relation represents y y as a function of x . x .

5 x + 2 y = 10 5 x + 2 y = 10

y = x 2 y = x 2

x = y 2 x = y 2

3 x 2 + y = 14 3 x 2 + y = 14

2 x + y 2 = 6 2 x + y 2 = 6

y = − 2 x 2 + 40 x y = − 2 x 2 + 40 x

y = 1 x y = 1 x

x = 3 y + 5 7 y − 1 x = 3 y + 5 7 y − 1

x = 1 − y 2 x = 1 − y 2

y = 3 x + 5 7 x − 1 y = 3 x + 5 7 x − 1

x 2 + y 2 = 9 x 2 + y 2 = 9

2 x y = 1 2 x y = 1

x = y 3 x = y 3

y = x 3 y = x 3

y = 1 − x 2 y = 1 − x 2

x = ± 1 − y x = ± 1 − y

y = ± 1 − x y = ± 1 − x

y 2 = x 2 y 2 = x 2

y 3 = x 2 y 3 = x 2

For the following exercises, evaluate f f at the indicated values f ( −3 ) , f ( 2 ) , f ( − a ) , − f ( a ) , f ( a + h ) . f ( −3 ) , f ( 2 ) , f ( − a ) , − f ( a ) , f ( a + h ) .

f ( x ) = 2 x − 5 f ( x ) = 2 x − 5

f ( x ) = − 5 x 2 + 2 x − 1 f ( x ) = − 5 x 2 + 2 x − 1

f ( x ) = 2 − x + 5 f ( x ) = 2 − x + 5

f ( x ) = 6 x − 1 5 x + 2 f ( x ) = 6 x − 1 5 x + 2

f ( x ) = | x − 1 | − | x + 1 | f ( x ) = | x − 1 | − | x + 1 |

Given the function g ( x ) = 5 − x 2 , g ( x ) = 5 − x 2 , evaluate g ( x + h ) − g ( x ) h , h ≠ 0. g ( x + h ) − g ( x ) h , h ≠ 0.

Given the function g ( x ) = x 2 + 2 x , g ( x ) = x 2 + 2 x , evaluate g ( x ) − g ( a ) x − a , x ≠ a . g ( x ) − g ( a ) x − a , x ≠ a .

Given the function k ( t ) = 2 t − 1 : k ( t ) = 2 t − 1 :

• ⓐ Evaluate k ( 2 ) . k ( 2 ) .
• ⓑ Solve k ( t ) = 7. k ( t ) = 7.

Given the function f ( x ) = 8 − 3 x : f ( x ) = 8 − 3 x :

• ⓐ Evaluate f ( − 2 ) . f ( − 2 ) .
• ⓑ Solve f ( x ) = − 1. f ( x ) = − 1.

Given the function p ( c ) = c 2 + c : p ( c ) = c 2 + c :

• ⓐ Evaluate p ( − 3 ) . p ( − 3 ) .
• ⓑ Solve p ( c ) = 2. p ( c ) = 2.

Given the function f ( x ) = x 2 − 3 x : f ( x ) = x 2 − 3 x :

• ⓐ Evaluate f ( 5 ) . f ( 5 ) .

Given the function f ( x ) = x + 2 : f ( x ) = x + 2 :

• ⓐ Evaluate f ( 7 ) . f ( 7 ) .

Consider the relationship 3 r + 2 t = 18. 3 r + 2 t = 18.

• ⓐ Write the relationship as a function r = f ( t ) . r = f ( t ) .
• ⓑ Evaluate f ( − 3 ) . f ( − 3 ) .
• ⓒ Solve f ( t ) = 2. f ( t ) = 2.

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

Given the following graph,

• ⓐ Evaluate f ( −1 ) . f ( −1 ) .
• ⓑ Solve for f ( x ) = 3. f ( x ) = 3.
• ⓐ Evaluate f ( 0 ) . f ( 0 ) .
• ⓑ Solve for f ( x ) = −3. f ( x ) = −3.
• ⓐ Evaluate f ( 4 ) . f ( 4 ) .
• ⓑ Solve for f ( x ) = 1. f ( x ) = 1.

For the following exercises, determine if the given graph is a one-to-one function.

{ ( −1 , −1 ) , ( −2 , −2 ) , ( −3 , −3 ) } { ( −1 , −1 ) , ( −2 , −2 ) , ( −3 , −3 ) }

{ ( 3 , 4 ) , ( 4 , 5 ) , ( 5 , 6 ) } { ( 3 , 4 ) , ( 4 , 5 ) , ( 5 , 6 ) }

{ ( 2 , 5 ) , ( 7 , 11 ) , ( 15 , 8 ) , ( 7 , 9 ) } { ( 2 , 5 ) , ( 7 , 11 ) , ( 15 , 8 ) , ( 7 , 9 ) }

For the following exercises, determine if the relation represented in table form represents y y as a function of x . x .

For the following exercises, use the function f f represented in Table 14 .

Evaluate f ( 3 ) . f ( 3 ) .

Solve f ( x ) = 1. f ( x ) = 1.

For the following exercises, evaluate the function f f at the values f ( − 2 ) , f ( − 1 ) , f ( 0 ) , f ( 1 ) , f ( − 2 ) , f ( − 1 ) , f ( 0 ) , f ( 1 ) , and f ( 2 ) . f ( 2 ) .

f ( x ) = 4 − 2 x f ( x ) = 4 − 2 x

f ( x ) = 8 − 3 x f ( x ) = 8 − 3 x

f ( x ) = 8 x 2 − 7 x + 3 f ( x ) = 8 x 2 − 7 x + 3

f ( x ) = 3 + x + 3 f ( x ) = 3 + x + 3

f ( x ) = x − 2 x + 3 f ( x ) = x − 2 x + 3

f ( x ) = 3 x f ( x ) = 3 x

For the following exercises, evaluate the expressions, given functions f , g , f , g , and h : h :

• f ( x ) = 3 x − 2 f ( x ) = 3 x − 2
• g ( x ) = 5 − x 2 g ( x ) = 5 − x 2
• h ( x ) = − 2 x 2 + 3 x − 1 h ( x ) = − 2 x 2 + 3 x − 1

3 f ( 1 ) − 4 g ( − 2 ) 3 f ( 1 ) − 4 g ( − 2 )

f ( 7 3 ) − h ( − 2 ) f ( 7 3 ) − h ( − 2 )

For the following exercises, graph y = x 2 y = x 2 on the given domain. Determine the corresponding range. Show each graph.

[ − 0.1 , 0.1 ] [ − 0.1 , 0.1 ]

[ − 10 ,  10 ] [ − 10 ,  10 ]

[ − 100 , 100 ] [ − 100 , 100 ]

For the following exercises, graph y = x 3 y = x 3 on the given domain. Determine the corresponding range. Show each graph.

[ − 0.1 ,  0 .1 ] [ − 0.1 ,  0 .1 ]

[ − 100 ,  100 ] [ − 100 ,  100 ]

For the following exercises, graph y = x y = x on the given domain. Determine the corresponding range. Show each graph.

[ 0 ,  0 .01 ] [ 0 ,  0 .01 ]

[ 0 ,  100 ] [ 0 ,  100 ]

[ 0 ,  10,000 ] [ 0 ,  10,000 ]

[ −0.001 , 0.001 ] [ −0.001 , 0.001 ]

[ −1000 , 1000 ] [ −1000 , 1000 ]

[ −1,000,000 , 1,000,000 ] [ −1,000,000 , 1,000,000 ]

Real-World Applications

The amount of garbage, G , G , produced by a city with population p p is given by G = f ( p ) . G = f ( p ) . G G is measured in tons per week, and p p is measured in thousands of people.

• ⓐ The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function f . f .
• ⓑ Explain the meaning of the statement f ( 5 ) = 2. f ( 5 ) = 2.

The number of cubic yards of dirt, D , D , needed to cover a garden with area a a square feet is given by D = g ( a ) . D = g ( a ) .

• ⓐ A garden with area 5000 ft 2 requires 50 yd 3 of dirt. Express this information in terms of the function g . g .
• ⓑ Explain the meaning of the statement g ( 100 ) = 1. g ( 100 ) = 1.

Let f ( t ) f ( t ) be the number of ducks in a lake t t years after 1990. Explain the meaning of each statement:

• ⓐ f ( 5 ) = 30 f ( 5 ) = 30
• ⓑ f ( 10 ) = 40 f ( 10 ) = 40

Let h ( t ) h ( t ) be the height above ground, in feet, of a rocket t t seconds after launching. Explain the meaning of each statement:

• h ( 1 ) = 200 h ( 1 ) = 200
• h ( 2 ) = 350 h ( 2 ) = 350

Show that the function f ( x ) = 3 ( x − 5 ) 2 + 7 f ( x ) = 3 ( x − 5 ) 2 + 7 is underline not end underline one-to-one.

• 1 http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014.
• 2 http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.

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Access for free at https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: Precalculus 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions
• Section URL: https://openstax.org/books/precalculus-2e/pages/1-1-functions-and-function-notation

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