Determine whether the graph is a graph of a function or not. (See Example 1.)Example 1Use...

Determine whether the graph is a graph of a function or not. (See Example 1.)

Example 1

Use the vertical-line test to determine which of the graphs in Figure 1 are the graphs of functions.

Solution

To use the vertical-line test, imagine dragging a ruler held vertically across the graph from left to right. If the graph is that of a function, the edge of the ruler would hit the graph only once for every x-value. If you do this for graph (a), every vertical line intersects the graph in at most one point, so this graph is the graph of a function. Many vertical lines (including the y-axis) intersect graph (b) twice, so it is not the graph of a function.

Graph (c) appears to fail the vertical-line test near x = 1 and x = −1, indicating that it is not the graph of a function. But this appearance is misleading because of the low resolution of the calculator screen. The table in Figure 2 and the very narrow segment of the graph in Figure 3 show that the graph actually rises as it moves to the right. The same thing happens near x = −1. So this graph does pass the vertical-line test and is the graph of a function. (It rule is f(x) = 15x11 − 2.) The moral of this story is that you can’t always trust images produced by a graphing calculator. When in doubt, try other viewing windows or a table to see what is really going on.

FIGURE 1

FIGURE 2

FIGURE 3