Mathematics 101 Transitional Mathematics

Study Guide :: Unit 3

Functions and Relations: Generally Speaking

Mathematical relations are not as different from human family relations as you might suppose. In mathematics, for example, relations are simply pairings of objects (mostly, but not always mathematical in nature) which are related to each other, generally in some well-defined way. Which comes first and which comes second in a pairing establishes an order on the pairings. Human relations are also pairings of individuals tethered together in some definable fashion. Who comes first and who comes second in human pairings is also very important when defining the relationship.

Consider (female, male) pairs where the relationship linking the two together is:

Marriage: “female is married to male.”

In mathematics, we would represent this relationship as follows:

M: { married females } → { males }

This relationship, called M for short, is a mathematical relation, as well as being indicative of the human relationship of marriage. The definition of the mathematical relation M, as it stands, has no additional restrictions on it. It does not preclude many females being married to a single male; nor does it forbid a single female having many different husbands. That is a true mathematical relation, a many-to-many pairing of objects where many means either greater than or equal to one for each pairing, not ‘many’ in the ordinary sense.

If restrictions are placed on the marriage relation M as to how many can be married to whom, then we get either monogamy, polygamy, or polyandry. The tutorials on this topic will teach you the various kinds of mathematical relations, their important properties, and which of them are ‘functional’.

Learning Objectives

After you have completed Unit 3, you should be able to:

  1. define, describe, and give examples of mathematical relations and mathematical functions, including what their domains, ranges, and values are in a given context;
  2. plot the graphs of functions and know what information can be deduced by looking at the geometric graphs of a function in a context;
  3. perform algebraic operations (addition, subtraction, multiplication, division, exponentiation, and composition) on various functions, including the application and recognition of both rigid and non-rigid transformations on them;
  4. define, describe, and give examples of one-to-one functions, including finding their inverses in both algebraic and graphical forms;
  5. explain the role of functions in modeling the real world, create mathematical models of real life situations using various kinds of functions, and solve practical problems using these models.

Readings and Practice

Chapter 3 (pp. 13–94) of the textbook

Online graphing tool: Graph go to link icon

What is a Function?

Textbook Readings

pp. 13–21

Practice

(pp. 21–24): 1–4, 5, 7, 15, 21, 25, 27, 31, 35, 49, 55, 59, 63, 65, 75, 79

Answers

p. 493

Terms to Understand

formal definitions of a mathematical relation and a mathematical function; function value or image; domain, target, and range sets of a function, independent variable, dependent variable, piecewise defined function

Figures/Tables

Four Ways to Represent a Function (p. 20)

Online Tutorials—the Overview

Pertinent to “What is a Function?”

  • Relations & Functions: an introduction go to link icon
  • Relations with associated definitions and examples go to link icon
  • Functions with associated definitions and examples go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Graphs of Functions

Textbook Readings

pp. 24–31

Optional Subsection

  • Graphing Functions with a Graphing Calculator (pp. 26–27)
Practice

(pp. 31–34): 1–4, 5, 9, 11, 15, 19, 23, 29 using Graph, 35, 39, 45, 51, 53, 57, 61, 65, 69, 81, 87

Answers

pp. 494–495

Terms to Understand

graph of a function, geometric graph of a function, Vertical Line Test

Figures/Tables

Categories of Functions (p. 31)

Online Tutorials—the Overview

Pertinent to “Graphs of Functions”

  • Functions with associated definitions and examples go to link icon
  • Examples of various kinds of functions, finding their values, their domains and ranges, their graphs, and whether they are one-to-one go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Getting Information from the Graph of a Function

Textbook Readings

pp. 35–40

Practice

(pp. 40–44): 1–4, 5, 9, 13, 15, 19, 21, 23 using Graph, 29 using Graph, 31, 33, 35, 41, 45, 51

Answers

pp. 495–496

Terms to Understand

graph of a function, geometric graph of a function, values of a function, domain and range of a function, increasing and decreasing functions, definitions of a local maximum value and a local minimum value of a function

Figures/Tables

Definition of Increasing and Decreasing Functions (p. 36)
Local Maxima and Minima of a Function (p. 38)

Online Tutorials—the Overview

Pertinent to “Getting Information from the Graph of a Function”

  • Functions with associated definitions and examples go to link icon
  • Examples of various kinds of functions, finding their values, their domains and ranges, their graphs, and whether they are one-to-one go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Average Rate of Change of a Function

Textbook Readings

pp. 44–48

Practice

(pp. 49–51): 1–4, 5, 7, 9, 13, 17, 23, 27, 31

Answers

p. 496

Terms to Understand

average rate of change of a function, average speed, constant rate of change

Figures/Tables

Average Rate of Change (p. 45)

Online Tutorials—the Overview

Pertinent to “Average Rate of Change of a Function”

  • Rates of Change as Slopes of Lines: with real world applications
    Rates of Change 1 go to link icon
    Rates of Change 2 go to link icon
    Rates of Change 3 go to link icon
    Rates of Change 4 go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Transformations of Functions

Textbook Readings

pp. 51–58

Practice

(pp. 59–62): 1–4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 29, 33, 45, 47, 53, 55, 63, 69 using Graph, 71 using Graph, 75, 77, 85, 87

Answers

pp. 496–498

Terms to Understand

rigid and a non-rigid transformation of a function, vertical and horizontal shifting of the graph of a function, reflections of the graph of a function in an axis, vertical and horizontal scaling of a function, definitions of even and odd functions

Figures/Tables

Vertical Shifts of Graphs (p. 52)
Horizontal Shifts of Graphs (p. 53)
Reflecting Graphs (p. 54)
Vertical Stretching and Shrinking of Graphs (p. 55)
Horizontal Stretching and Shrinking of Graphs (p. 56)
Even and Odd Functions (p. 58)

Online Tutorials—the Overview

Pertinent to “Transformations of Functions”

  • Rigid transformations of functions: translations and reflections go to link icon
  • Non-rigid transformations of functions: expansions and contractions go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Combining Functions

Textbook Readings

pp. 62–67

Practice

(pp. 68–70): 1–4, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 33, 35, 37, 45, 59, 63, 67

Answers

pp. 498–499

Terms to Understand

the definitions of sum, difference, product, quotient, and composition of functions

Figures/Tables

Algebra of Functions (p. 63)
Composition of Functions (p. 65)

Online Tutorials—the Overview

Pertinent to “Combining Functions”

  • Operations on Functions: addition, subtraction, multiplication, division and composition go to link icon
  • Composing transformations go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

One-to-One Functions and their Inverses

Textbook Readings

pp. 71–76

Practice

(pp. 76–79): 1–4, 5, 7, 9, 11, 13, 17, 21, 25, 29, 31, 37, 41, 43, 51, 57, 59, 61, 65 using Graph, 67 using Graph, 73 using Graph, 75, 81, 83, 85, 89, 91 a, b

Answers

pp. 499–500

Terms to Understand

definition of a one-to-one function, Horizontal Line Test, definition of the inverse of a function, properties of inverse functions

Figures/Tables

Definition of One-to-One Function (p. 71)
Horizontal Line Test (p. 71)
Definition of the Inverse of a Function (p. 73)
Inverse Function Property (p. 73)
How to Find the Inverse of a One-to-One Function (p. 74)

Online Tutorials—the Overview

Pertinent to “One-to-One Functions and their Inverses”

  • Examples of various kinds of functions, finding their values, their domains and ranges, their graphs, and whether they are one-to-one go to link icon
  • Functions and Their Inverses: for the operations of addition, multiplication and composition go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Focus on Modeling: Modeling with Functions

Textbook Readings

pp. 85–90

Practice

(pp. 90–94): 1, 3, 5, 7, 9, 11, 13, 21, 27

Answers

p. 502

Terms to Understand

mathematical model, definition of the inverse of a function, properties of inverse functions

Figures/Tables

Guidelines for Modeling with Functions (p. 87)

Online Tutorials—the Overview

Pertinent to “Focus on Modeling: Modeling with Functions”

  • Translation, Analysis, and Solution of Real World Problems go to link icon
  • Rates of Change as Slopes of Lines: with real world applications
    Rates of Change 2 go to link icon
    Rates of Change 3 go to link icon
    Rates of Change 4 go to link icon

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Unit 3 Review

(pp. 79–82): odd numbers

Answers

pp. 500–501

Unit 3 Test

(pp. 83–84): 1–12

Answers

pp. 501–502

Figures/Tables

Formulas to Remember (pp. iv and v at the beginning of the text)