2.6 Domain and Range Using Graph

Drawing the graph of a function is a visual way of identifying its domain and often the most convenient way for determining its range.To figure out if a specific value is in the range of the function, we (imaginarily) draw a horizontal line passing through that specific $y$ -value. If that line intersects the graph, then that $y$ -value is in the range of the function. This means the horizontal projection of the graph onto the $y$ -axis is the range of the function.

Similarly, to find out if a specific $x$ -value is in the domain of the function, we draw a vertical line passing through that value of $x$ . If this vertical line intersects the graph, then that specific value of $x$ is in the domain of the function. In other words, the domain of a function is the vertical projection of its graph onto the $x$ -axis. The procedure for finding the domain and the range using the graph is shown in Figure 1.

Figure 1: Finding the domain and the range of a function using its graph

Example 1

The functions

g (x)

and

ϕ (x)

are graphed below. Find their domains and ranges.

Figure 2

Solution

Any vertical line intersects the graph of $g (x)$ except when $x = 1$ .This means $x = 1$ is not in the domain of $g$ . Horizontal lines hit its graph only when $y = 1$ or $y = 2$ . Therefore

$D o m (g) = {x | x \neq 1}, a n d R n g (g) = {1, 2} .$

The domain of $g$ can also be rewritten as

$D o m (g) = R - {1}$ or $D o m (g) = (- \infty, 1) \cup (1, \infty) .$

For $ϕ (x)$ , any vertical line intersects the graph. Although a vertical line passing through $x = 1$ does not intersect the right piece of the graph, it meets the left piece. Therefore, all real numbers are in the domain of $ϕ$ . Any horizontal line passing through $y \geq 0$ or $y = 1$ intersects the graph. Thus,

$D o m (ϕ) = R$ and $R n g (ϕ) = {x | x \geq 0 or x = - 1} = [0, \infty) \cup {- 1} .$

Example 2

Find the range of

f

f : [- 1, 2] \to R, a n d f (x) = x^{2} - 3.

Solution

To find the range of

f

, we first sketch its graph by constructing a table of

(x, f (x))

values, plotting these points, and then connecting them. We make sure that the end values of the interval

[- 1, 2]

are among the values that we choose for

x

in our table:

The graph of $f$ is shown in Figure 3. If we look at this graph, we realize that the function never goes below $- 3$ and beyond 1. That is, $R n g (f) = [- 3, 1]$ . We will learn how to easily graph such a function later in this chapter.

Figure 3. Graph of

y = x^{2} - 3

when

- 1 \leq x \leq 2