Domain and Range

Example 1:

Consider the function shown in the diagram.

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

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

Example 2:

Consider the relation $\left\{\left(0,7\right),\left(0,8\right),\left(1,7\right),\left(1,8\right),\left(1,9\right),\left(2,10\right)\right\}$ .

Here, the relation is given as a set of ordered pairs. The domain is the set of $x$ -coordinates, $\left\{0,1,2\right\}$ , and the range is the set of $y$ -coordinates, $\left\{7,8,9,10\right\}$ . Note that the domain elements $1$ and $2$ are associated with more than one range elements, so this is not a function.

Example 3:

The domain of the function

$f\left(x\right)=\frac{1}{x}$

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 $y=0$ .

Example 4:

The following notation shows that the domain of the function is restricted to the interval $\left(-1,1\right)$ .

$f\left(x\right)=x^2,-1<x<1$

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

$0\le y<1$ .