The **domain** of a function *f*(*x*) is the set of all values for which the function is defined, and the **range** of the function is the set of all values that *f *takes.

(In grammar school, you probably called the domain the replacement set and the range the solution set. They may also have been called the input and output of the function.)

**Example 1:**

Consider the function shown in the diagram.

Here, the domain is the set {*A*, *B*, *C*, *E*}. *D* is not in the domain, since the function is not defined for *D*.

The range is the set {1, 3, 4}. 2 is not in the range, since there is no letter in the domain that gets mapped to 2.

**Example 2:**

The domain of the function

* f*(

is all real numbers except zero (since at *x* = 0, the function is undefined: division by zero is not allowed!).

The range is also all real numbers except zero. You can see that there is some point on the curve for every *y*-value except *y* = 0.

Domains can also be explicitly specified, if there are values for which the function could be defined, but which we don't want to consider for some reason.

**Example 3:**

The following notation shows that the domain of the function is restricted to the interval (–1, 1).

*f*(*x*) = *x*^{2}, –1 *x* 1

The graph of this function is as shown. Note the open circles, which show that the function is not defined at *x* = –1 and *x* = 1. The *y*-values range from 0 up to 1 (including 0, but not including 1). So the range of the function is

0 *y* < 1.