Domain and Range of a Function

You have seen that a function is a rule that assigns to each input number, x, exactly one output number, y.

The set of x-values is called the domain of the function.

The set of y-values is called the range of the function.

Example 1

Given the function: y = x²

a. Find the domain.

b. Find the range.

Solution

a. To find the domain, ask yourself, “What is x allowed to be?”. Since we can square any real number, the domain of y = x² is all real numbers.

b. To find the range, ask yourself, “What results when we square a real number?” Squaring a real number always results in 0 or a positive real number. Thus, the range of y = x² is y ≥ 0.

We can see from the graph that the smallest value of y is 0.

The domain and range of a function can be expressed in several different ways.

Represent the domain of the function shown in the graph:

a. Using words.

b. Using inequality symbols.

c. Using a number line.

d. Using interval notation.

Solution

The function consists of all the ordered pairs on the line. The x-values start at -6 and go up to, but do not include, 4.

a. The domain is all real numbers between -6 and 4, including -6.

Or, we could say “the domain is all real numbers greater than or equal to -6 and less than 4.”

b. -6 ≤ x < 4

c.

Note:

Remember, a closed circle, , on a graph means a point is included in the solution; an open circle, , means the point is not included.

d. Interval notation indicates the domain by stating the end points of an interval.

If an end point is included, we use a square bracket, [ or ]; if the end point is not included, we use a parenthesis, ( or ). For the function shown, we write the domain as [-6, 4):

• the domain starts at and includes -6, so we use a [ next to -6.

• the domain ends at, but does not include, 4, so we use a ) next to 4.

Note:

Interval notation can look like an ordered pair. For example, (2, 5) can have two different meanings depending on the context in which it is used:

• If we are talking about a domain, then the interval (2, 5) represents all the values between 2 and 5 (not including 2 or 5).

• If we are talking about a point on the xyplane, then the ordered pair (2, 5) represents the coordinates of a point. The x-coordinate is 2 and the y-coordinate is 5.