College Algebra Chapter 2 Functions and Graphs Section 2.3 Functions and Relations Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Determine Whether a Relation is a Function Apply Function Notation Determine x- and y-intercepts of a Function Defined by y = f(x) Determine Domain and Range of a Function Interpret a Function Graphically

Concept 1 Determine Whether a Relation is a Function

Determine Whether a Relation is a Function (1 of 2) Definition of a Relation: A set of ordered pairs (x, y) is called a relation in x and y. The set of x values in the ordered pairs is called the domain of the relation. The set of y values in the ordered pairs is called the range of the relation.

Determine Whether a Relation is a Function (2 of 2) Definition of a Function: Given a relation in x and y, we say that y is a function of x if for each value of x in the domain, there is exactly one value of y in the range.

Skill Practice 1 For the table shown. Write the set of ordered pairs that defines the relation. Write the domain. Write the range. X 3 -2 5 1 y -4

Examples 1, 2 Determine if the relation defines y as a function of x Solution: {(2,3)}, {(2,4)}, {(3,5)} {(2,3)},{(2,4)} → diff. y same x No, not a function. {(1,3)}, {(2,3)}, {(3,5)} Yes

Skill Practice 2 Determine if the relation defines y as a function of x. {(8,4), (3,-1), (5,4)} {(-3,2), (9,5), (1,0), (-3,1)}

Example 3 Determine if the relation defines y as a function of x Solution: Yes

Examples 4, 5 Determine if the relation defines y as a function of x y = 2x + 6 Solution: Yes Solution: Yes

Examples 6 – 8 (1 of 2) Determine if the relation defines y as a function of x x = 7- y Solution: Y=7-x Yes

Examples 6 – 8 (2 of 2)

Determine Whether a Relation is a Function Vertical Line Test: A graph defines y as a function of x if no vertical line intersects the graph in more than one point.

Examples 9 – 11 (1 of 3) Determine if the graph defines y as a function of x Solution: No

Examples 9 – 11 (2 of 3) yes

Examples 9 – 11 (3 of 3) Yes

Skill Practice 3 Determine if the given relation defines y as function of x.

Skill Practice 4 Determine if the relation defines y as a function of x. |y+1|

Concept 2 Apply Function Notation

Example 12 (1 of 2)

Example 12 (2 of 2)

Example 13 Solution:

Skill Practice 5 Evaluate the function defined by h(x)=4x-3 for the given values of x. h(-3) h(-1) h(0) h(1) h(3)

Skill Practice 6 Evaluate the function defined by for the given values of x. f(t) f(a+h)

Concept 3 Determine x- and y-intercepts of a Function Defined by y = f(x)

Determine x- and y-intercepts of a Function Defined by y = f(x) Given a function defined by y = f (x): The x-intercepts are the real solutions to the equation f (x) = 0. The y-intercept is given by f (0).

Example 14 Find the x-and y-intercepts. Solution:

Example 15 Find the x-and y-intercepts. g(x) = 3 - |x| Solution:

Example 16 Find the x-and y-intercepts. Solution:

Skill Practice 7 Find the x-and y-intercepts of the function defined by f(x) =|x|-5.

Concept 4 Determine Domain and Range of a Function

Determine Domain and Range of a Function Given a relation defining y as a function of x, the domain is the set of x values in the function, and the range is the set of y values in the function.

Example 17 Determine the domain and range for the function Solution:

Example 18 Determine the domain and range for the function Solution:

Example 19 Determine the domain and range for the function Solution:

Example 20 Determine the domain and range for the function Solution:

Skill Practice 8 Determine the domain and range for the functions shown.

Skill Practice 9 Write the domain of each function in interval notation.

Concept 5 Interpret a Function Graphically

Example 21 (1 of 2) Use the graph of y=f(x) to answer the following: Solution: f(-3)=-3 f(0)=2 find all x for which f(x)=-1 y=-1 x=-1

Example 21 (2 of 2) Find all x for which f(x) f(x)=1 y=-1 x=2 Determine the x-intercept. No x-integer Determine the y-intercept. (0,2) Determine the domain of f Determine the range of f.

Skill Practice 10 (1 of 2) Use the function f pictured to find: f(-2) All x for which f(x) =3. All x for which f(x)=1. The x-intercept(s). The y-intercept. The domain of f. The range of f.

Skill Practice 10 (2 of 2)