# 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 = x^{2}

a. Find the domain.

b. Find the range.

**Solution **

a. To find the domain, ask yourself, â€œWhat is x allowed to be?â€.
Since we can square any real number, the domain of y = x^{2} is all real
numbers.

b. To find the range, ask yourself, â€œWhat results when we square a real
number?â€ Squaring a real number always results in 0 or a positive real
number. Thus, the range of y = x^{2} 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 â€œthe domain is all real numbers greater than or equal
to -6 and less than 4.â€

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):

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

â€¢ 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:

â€¢ 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).

â€¢ 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.