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, “What is x allowed to be?”. The square root of a number results in a real number only if the radicand is greater than or equal to zero.

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 y-values for the x-values 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 “positive infinity.” 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]

“All real numbers” 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, “What is x allowed to be?”.

We can divide by any number except 0.

So, the domain is all real numbers except 5. We write this as “all real numbers where x 5”. 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 “all real numbers where y ≠ 0” or as (-∞, 0) (0, +∞).

Note:

We can write