DOMAIN AND RANGE
A function is a set of ordered pairs(x,y). The set of all possible x-values (abscissa) is called the
domain of the function while the set of all possible y-values (ordinate) is called the range of a
function.
First, let’s determine how to identify the domain and range of a given ordered pair.
Example 1: Determine the domain and the range of the following functions
a. B= {(1,2), (-4,3), (2,-3), (0,0)}
b. C = {(-9,1), (3,1), (-5,1), (4,1)}
c.
x 3 5 7 8 -2
y 3 4 7 9 -1
d.
Solution:
a. Domain = { -4,0 ,1,2} c. Domain = { -2,3,5,7,8}
Range = { -3,0,2,3 } Range = { -1,3,4,7,9 }
b. Domain = {-9,-5,3,4} d. Domain = {5,6,7}
Range = {1} Range = {1,2,4}
We learned how to identify the domain and range of an ordered pair based on Example 1. Now
let’s use what we learned to identify the domain and range of certain functions.
Example 2: Find the domain and range of the following function y = f(x)
a. Y = 3x + 2 c. y = 2 x 2
b. Y = -3 x 3-4 d. y = │-3x│ e. y = 5
Solution:
a. This is a polynomial function. Since polynomial functions are defined or exist in
any real number, the domain is { x / x is a real number} and the range is { y / y is a
real number}.
b. This also a polynomial function of third degree. Thus the domain and the range of
function are the set of real numbers.
c. The domain is the set of real numbers. Notice that the value of y will never be
negative. Therefore, the range is the set of all real numbers greater or equal to zero.
d. The domain is a real number. Since the absolute value of any real number will
never be negative, then the range is {y / y≥0}.
e. F(x) = 5 is an example of a constant function. The value of a function is always
equal to 5 for any value of the independent variable x. Consequently, the domain is
{x / x is a real number} and the range is {5}.
Example3: Determine the domain and range of the given functions.
5 x+3
a. f(x) = b. y = c. g(x) = √ x 2−25
x+2 x−2
Solution:
a. Note that x + 2 is 0 when x = -2, which makes the function undefined. Therefore, the domain
is the set of real numbers not equal to -2. Observe further that the function values will never be
equal to 0. Thus, the range is the set of real numbers not equal to 0.
b. The domain is the set of real numbers not equal to 2. To find the range, solve for x in the
given equation.
x+3
y=
x−2
The resulting equation suggests that x is defined for
y(x-2) = x + 3
all real numbers except 1. Therefore, the range is the
xy – 2y = x + 3 set of real numbers not equal to 1.
xy – x = 2y + 3
x(y – 1) = 2y + 3
2 y +3
x= y−1
c. The function g(x) = √ x 2−25 is defined only when x 2−25 ≥ 0 or (x-5)(x+5) ≥0. Solving for x,
case 1: x- 5 ≥0 and x+ 5 ≥ 0 case 2: x- 5 ≤0 and x+ 5 ≤ 0
x ≥5 and x ≥ -5 x≤ 5 and x ≤ -5
For every value of x satisfying either of these inequalities, a value of the function is determined.
If we take values of x between -5 and 5, the expression x 2−25 becomes negative which makes
g(x) undefined. These verify that the domain of g is (-∞ ,5] ∪ [5,∞ ) which is shown in the
above solution, and the range is {y/y ≥0}.
There are functions which are piecewise in nature. These are functions which are defined in
different domains since they are determined by several equations.
Example 3: Determine the domain and range of the following piecewise functions.
a. f(x) = {2 x4+3if x=2
if x ≠ 2
b. g(x) = {−x2 x+3+2ifif xx<1≥1
2
Solution:
a. The function f(x) = 2x + 3 is a linear function. Hence, it contains all ordered pairs of real
numbers satisfying this equation, except (2,7), since 2 is not included in the domain. However,
the function is equal to 4 when x is 2. Consequently, the domain of f is the set of real numbers,
and the range is the set of real numbers ≠ 7.
b. The table of values for each function is given below.
1. g(x) = 2x + 3 for x< 1
x -1 0 1 3 0.8 0.9 0.99
2 4
y 1 3 4 4.5 4.6 4.8 4.98
2. g(x) = - x 2 + 2 for x ≥1
x 1 2 4 8 16
y 1 -2 -14 -62 -254
Since x< 1 in the first equation for g, observe that x gets closer to 1, the value of g gets closer
to 5 but will never be equal to 5. Hence, the range is the set of real numbers less than 5. In the
second equation for g, the range is the set of real number less than or equal to 1. Combining the
results, the domain of g is the set of real numbers, and the range is {y/y < 5}.
