186 CHAPTER 3 Graphs, Linear Equations, and Functions
Agreement on Domain
Unless specified otherwise, the domain of a relation is assumed to be all real num-
bers that produce real numbers when substituted for the independent variable.
NOTE Graphs that do not represent functions are still relations. All equations and
graphs represent relations, and all relations have a domain and range.
Relations are often defined by equations. If a relation is defined by an equation,
keep the following guidelines in mind when finding its domain.
1. Exclude from the domain any values that make the denominator of a fraction
equal to 0.
Example:The function defined by has all real numbers except 0 as its
domain, since division by 0 is undefined.
2. Exclude from the domain any values that result in an even root of a negative
number.
Example:The function defined by has all nonnegativereal numbers as
its domain, since the square root of a negative number is not real.
In this book, we assume the following agreement on the domain of a relation.
y= 2 x
y=
1
x
Identifying Functions from Their Equations
Decide whether each relation defines yas a function of x, and give the domain.
(a)
In the defining equation (or rule) yis always found by adding 4 to x.
Thus, each value of xcorresponds to just one value of y, and the relation defines a
function. Since xcan be any real number, the domain is
or
(b)
For any choice of xin the domain, there is exactly one corresponding value for y
(the radical is a nonnegative number), so this equation defines a function. Since the
equation involves a square root, the quantity under the radical symbol cannot be
negative — that is, must be greater than or equal to0.
Add 1.
Divide by 2.
The domain of the function is
(c)
The ordered pairs and both satisfy this equation. Since one
value of x, 16, corresponds to two values of y, 4 and , this equation does not
define a function. Because xis equal to the square of y, the values of xmust always
be nonnegative. The domain of the relation is 3 0, q 2.
- 4
1 16, 4 2 1 16, - 42
y^2 = x
C
1
2 , qB.
xÚ
1
2
2 xÚ 1
2 x- 1 Ú 0
2 x- 1
y= 22 x- 1
5 x|x is a real number 6 , 1 - q, q 2.
y= x+ 4,
y=x+ 4
EXAMPLE 5
