Domain and Range of a Function
Example 1
Given the function:
a. Find the domain.
b. Find the range.
Solution
a. To find the domain, ask yourself, â€œWhat is x allowed to be?â€. The
square root of a number results in a real number only if the
radicand is greater than or equal to zero.
Thus, x  3 must be greater than or equal to 0.
Add 3 to both sides. 
x  3 ≥ 0
x ≥ 3 
So, we can write the domain of
as x
≥ 3.
Using interval notation, this is written [3, +∞).
b. To determine the range, find all possible yvalues for the xvalues in
the domain.
Taking a square root always results in a nonnegative number. Thus, the
smallest
can be is 0 (that happens when x
= 3).
When x is larger than 3,
is a positive number.
So, the range of y =
is y
≥ 0.
Note:
The symbol +∞ means â€œpositive infinity.â€
Since infinity is greater than any real
number, we must use a parenthesis when
using ∞ in interval notation.
For example:
x > 5 in interval notation is (5, +∞)
x ≤ 7 in interval notation is (∞, 7]
â€œAll real numbersâ€ can be expressed in
interval notation as (∞, +∞ ).
Example 2
Given the function:
a. Find the domain.
b. Find the range.
Solution
a. To find the domain, ask yourself, â€œWhat is x allowed to be?â€.
We can divide by any number except 0.
Thus, we must determine when the denominator is 0.
Add x to both sides. 
5  x = 0
5 = x 
So, the domain is all real numbers except 5. We write this as â€œall real
numbers where x
5â€. We can also write the domain as
(∞, 5)
(5,
+∞ ). The symbol,
, means
union. The union of two
sets is all the elements in both sets combined.
Notice that the graph gets closer and closer to the vertical line x = 5, but never touches it. We call the line x
= 5 an asymptote. An
asymptote is a line that the function approaches but never reaches.
b. To find the range, note that no matter how large x gets in
the
value of this fraction will never be 0. We can also see this from the
graph. The line y = 0 is an asymptote.
So, the range of y =
is all real numbers except 0.
We can write the range as â€œall real numbers where y
≠ 0â€ or as (∞, 0)
(0,
+∞).
Note:
We can write
