Chapter 3 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Introduction to Functions Understand the definition of a relation. Understand the definition of a function. Decide whether an equation defines a function. Find domains and ranges. Use function notation. Apply the function concept in an application.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Understand the definition of a relation. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Understand the definition of a relation. In an ordered pair (x, y), x and y are called the components of the ordered pair. Slide Any set of ordered pairs is called a relation. The set of all first components of the ordered pairs of a relation is the domain of the relation, and the set of all second components of the ordered pairs is the range of the relation.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solution: Identifying Domains and Ranges of Relations Defined by Ordered Pairs Use ordered pairs to define the relation. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Understand the definition of a function. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Understand the definition of a function. A very important type of relation called a function. A function is a set of ordered pairs in which each first component corresponds to exactly one second component. Slide By definition, the relation in the following order pairs is not a function, because the same first component, 3, corresponds to more then one second component. If the ordered pairs from this example were interchanged, giving the relation the result would be a function. In that case, each domain (first component) corresponds to exactly one range element (second component).

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Solution: function Determining Whether Relations Are Functions Determine whether each relation is a function. Slide Solution: not a function

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Decide whether an equation defines a function. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Decide whether an equation defines a function. Slide Given the graph of an equation, the definition of a function can be used to decide whether or not the graph represents a function. By the definition of a function, each x-value must lead to exactly one y-value. The way to determine if a graph is a function is the vertical line test. If a vertical line intersects a graph in more than one point, then the graph is not the graph of a function. Any nonvertical line is the graph of a function. For this reason, any linear equation of the form y = mx + b defines a function. (Recall that a vertical line has an undefined slope.)

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Deciding Whether Relations Define Functions Determine whether each relation is a function. Slide Solution: functionSolution: not a function

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Find domains and ranges. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley By the definitions of domain and range given for relations, the set of all numbers that can be used as replacements for x in a function is the domain of the function. The set of all possible values of y is the range of the function. Slide Find domains and ranges.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Find the domain and range of the function y = x Solution: Domain: Finding the Domain and Range of Functions Slide Range:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Use function notation. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use function notation. The letters f, g, and h are commonly used to name functions. For example, the function y = 3x + 5 may be written where f (x) is read “f of x.” The notation f (x) is another way of writing y in a function. For the function defined by f (x) = 3x + 5, if x = 7, then Read this result, f (7) = 26, as “f of 7 equals 26.” The notation f (7) means the values of y when x is 7. The statement f (7) = 26 says that the value of y = 26 when x is 7. It also indicates that the point (7,26) lies on the graph of f. Slide The notation f (x) does not mean f times x ; f ( x ) means the value of x for the function f. It represents the y –value that corresponds to x.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Notation In the notation f(x), fis the name of the function, xis the domain value, andf(x)is the range value y for the domain value x. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solution: Using Function Notation For the function f (x) = 6x − 2, find f (−1). Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 Objective 6 Apply the function concept in an application. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Applying the Function Concept to Population The number of U.S. students ages 3 – 21served by educational programs for students with disabilities for selected years are given in the table. Slide a) Write a set of ordered pairs that defines a function f for these data. b) Give the domain and range of f. c) Find f (1998). d) In which year did the number of students equal 5.7 million? That is, for what value of x does f (x) = 5.7million?

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Applying the Function Concept to Population (cont’d) Solution: The number of U.S. students ages 3 – 21served by educational programs for students with disabilities for selected years are given in the table. Slide a) b) c) d)