Mathematics 209: Finite Mathematics

Study Guide

Unit 2: Functions and Graphs

In this unit, you will learn to apply different types of functions to solve practical problems in economics, business, social sciences and life sciences. It is our hope that in the process you will come to appreciate how powerful the concept of function is in problem-solving.

Objectives

When you have completed this unit, you should be able to

1. apply the definition of a function to practical situations.
2. apply the vertical line test to determine whether a given curve corresponds to a function.
3. find the domain and range of a function.
4. sketch the graphs of elementary functions.
5. apply basic transformations (translation, expansion, contraction, reflection) to sketch graphs of functions.
6. identify the vertex and axis of symmetry of the graph of a quadratic function.
7. calculate break-even points using linear and quadratic functions to model cost, revenue, and profit.
8. identify polynomial and rational functions.
9. calculate compounded interest.
10. apply exponential functions to solve problems of exponential growth.
11. apply exponential functions to solve problems of decay.
12. compute logarithmic regression models for given data sets.

Functions

Indications

1. Read Section 2-1, Functions, on pages 47–59 of the textbook.
2. Do odd-numbered exercises 1–45, 55–77 and 79–87 on pages 59–61. If you have difficulty, consult your tutor to discuss the problem.
3. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

1. A function is a correspondence between two variables, where one depends on the other. Identify the dependent variable in the list of correspondences given on page 49 of the textbook in such a way as to make the correspondence a function.
2. When do we use the vertical line test?
3. What is the meaning of the notation $f\left(x\right)$?
4. If $p=m-nx$ is a price-demand function, what is $m\text{?}$ what is $x\text{?}$
5. If $P$ is a profit function, and $P\left(a\right)=0\text{,}$ then what is $a\text{?}$

Elementary Functions: Graphs and Transformations

Indications

1. Read Section 2–2, Elementary Functions: Graphs and Transformations, on pages 63–72 of the textbook.
2. Do odd-numbered exercises 1–59 on pages 73–74 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd numbered problems 61–69 on pages 74–75 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

6. Why, if ${\displaystyle f\left(x\right)=\sqrt{x}\text{,}}$ does ${\displaystyle f\left(\frac{-1}{2}\right)}$ not exist?
7. Why is the domain of the function ${\displaystyle f\left(x\right)=\sqrt{\left|x\right|}}$ all real numbers?
8. What transformation would we apply to the function $f\left(x\right)$ so that the graph of the resulting function $g\left(x\right)$ is equal to the graph of $f\text{,}$ but shifted 5 units up, and stretched vertically 2 units, in that order?
9. Why do we represent the graph of a function $f\left(x\right)$ on the Cartesian plane?

Quadratic Functions

Indications

1. Read Section 2-3, Quadratic Functions, on pages 76-90 of the textbook.
2. Do odd numbered exercises 1–51 on pages 90–91 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve problems 53–63 on pages 91–93 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

10. How do we know whether the parabola $f\left(x\right)=a{x}^2+bx+c$ opens upwards or downwards?
11. How do we know whether the parabola $f\left(x\right)=a{x}^2+bx+c$ does or does not cross the $x$-axis?
12. Are there rational functions whose domain is all real numbers?
13. Is a polynomial function a rational function?

Exponential Functions

Indications

1. Read Section 2–4, Exponential Functions, on pages 93–102 of the textbook.
2. Do odd numbered exercises 1–33 and 39–55 on pages 102–103 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd-numbered problems 61–77 on pages 104–105 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

14. For which values of $a$ is the exponential function $a^x$ decreasing?
15. Why is the base $b$ of an exponential function $b^x$ positive?
16. Is it true that if $e^x=e^y\text{,}$ then $x=y\text{?}$ Why?
17. Why is compound growth an application of exponential functions?
18. What is the function or mathematical model that best fits a data set of an exponential regression?

Logarithmic Functions

Indications

1. Read Section 2–5, Logarithmic Functions, on pages 105–116 of the textbook.
2. Do odd-numbered exercises 1–41 and 53–83, on pages 116–117 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
3. Solve odd numbered problems 93–101 on pages 117–118 of the textbook. If you have difficulty, consult your tutor to discuss the problem.
4. Answer the questions listed below, and then compare your answers with those given in the Answers to Study Guide Questions.

Questions

19. Why are the exponential and logarithmic functions invertible?
20. Why does ${\displaystyle {log}_{64}8=\frac{1}{2}}$?
21. Is it true that if $lnx=lny\text{,}$ then $x=y\text{?}$ Why?
22. Why are the properties of the logarithmic functions “dual” to the properties of the exponential functions?
23. Why is it that any exponential function can be represented as an exponential function in base $e\text{?}$
24. Why is it that any logarithmic function can be represented as a natural logarithmic function?

Finishing This Unit

1. Review the objectives of this unit and make sure you are able to meet each of them.
2. Study the section of the Chapter 2 Review titled Important Terms, Symbols and Concepts, on pages 119-121 of the textbook.
3. If there is a concept, definition, example or exercise that is not yet clear to you, go back and re-read it. Contact your tutor if you need help.
4. You may want to do problems 1–3, 5, 7, 10, 12, 14, 22, 25, 27, 29, 33, 39, 43, 44, 49, 58, 62, 75, 77, 82, 85, 90, 93, 95, 97 and 101 from the Review Exercise section on pages 121–125 of the textbook. The questions on the practice examination are taken from this exercise. If you have difficulty, consult your tutor to discuss the problem.
5. Complete the practice examination provided for this unit. Evaluate yourself, first checking your answers against those provided in the Answers section at the end of the textbook, and then comparing your solutions with those provided on pages S-70 to S-85 of the Student Solutions Manual portion of the textbook. The number of points in a question may indicate the number of steps in the solution. Give yourself full credit if your answer is correct and you give a complete solution, even if your solution differs from that shown in the Student Solutions Manual.
6. Once you have written and evaluated the practice examination, complete and submit the first assignment. You will find instructions in the Assignment Drop Box on the course home page.

Practice Examination

Time: 2 hours
Total points: 54
Passing grade: 50%

Do the following exercises from the Chapter 2 Review Exercise on pages 121–125 of the textbook.

To obtain full credit you must justify all your answers and show your work.

1. Exercise 22  (Marks: 2/2/2/2 for a total of 8 pts.)
2. Exercise 24  (Marks: 2/2 for a total of 4 pts.)
3. Exercise 44  (Marks: 4 pts.)
4. Exercise 60  (Marks: 4 pts.)
5. Exercise 76  (Marks: 10 pts.)
6. Exercise 84  (Marks: 6 pts.)
7. Exercise 96  (Marks: 6 pts.)
8. Exercise 100  (Marks: 12 pts.)