Intermediate Algebra (11th edition)

182 CHAPTER 3 Graphs, Linear Equations, and Functions

Since paycheck amount dependson number of hours worked, paycheck amount is

called the dependent variable,and number of hours worked is called the independent

variable.Generalizing, if the value of the variable ydepends on the value of the vari-

able x, then yis the dependent variableand xis the independent variable.

Independent variable Dependent variable

1 x, y 2

OBJECTIVE 2 Define and identify relations and functions. Since we can

write related quantities as ordered pairs, a set of ordered pairs such as

is called a relation.

51 5, 40 2 , 1 10, 80 2 , 1 20, 160 2 , 1 40, 320 26

Relation

A relationis any set of ordered pairs.

Afunctionis a special kind of relation.

Function

A functionis a relation in which, for each value of the first component of the

ordered pairs, there is exactly one valueof the second component.

NOW TRY
EXERCISE 1
Determine whether each rela-
tion defines a function.

(a)

(b) 51 - 1, - 32 , 1 0, 2 2 , 1 - 1, 6 26

51 1, 5 2 , 1 3, 5 2 , 1 5, 5 26

Determining Whether Relations Are Functions

Determine whether each relation defines a function.

(a)

For there is only one value of

For there is only one value of

For there is only one value of

Thus, relation Fis a function, because for each different x-value, there is exactly

one y-value.

(b)

Relation Gis also a function. Although the last two ordered pairs have the same

y-value (1 is paired with 2 and 2 is paired with 2), this does not violate the definition

of a function. The first components (x-values) are different, and each is paired with

only one second component (y-value).

(c)

In relation H, the last two ordered pairs have the same x-value paired with two dif-

ferent y-values ( is paired with both 1 and 0), so His a relation, but nota function.

Different y-values

Not a function

Same x-value

H= 51 - 4, 1 2 , 1 - 2 , 12 , 1 - 2 , 026

- 2

H= 51 - 4, 1 2 , 1 - 2, 1 2 , 1 - 2, 0 26

G= 51 - 2, - 12 , 1 - 1, 0 2 , 1 0, 1 2 , 1 1, 2 2 , 1 2, 2 26

x= 3 , y, - 1.

x= - 2 , y, 4.

x= 1 , y, 2.

F= 511 , 22 , 1 - 2 , 42 , 13 , - 126

EXAMPLE 1

In a function, no two ordered pairs can have the same first component and different

second components. NOW TRY

NOW TRY ANSWERS

1. (a)yes (b)no