Determine whether each of these functions is one-to-one:

Solution (a) Figure 49 illustrates the Horizontal-line Test for the graph of $f(x) = x^{2}$. The horizontal line $y = 1$ intersects the graph of $f$ twice, at $(1, 1)$ and at $(-1, 1)$, so $f$ is not one-to-one.

Figure 49

(b) Figure 50 illustrates the horizontal-line test for the graph of $g(x) = x^{3}$. Because every horizontal line intersects the graph of $g$ exactly once, it follows that $g$ is one-to-one.

Figure 50