The problem asks whether the function $g(x) = x^3$, whose graph is provided, has an inverse.

1. Problem Description

The problem asks whether the function $g(x) = x^3$, whose graph is provided, has an inverse.

2. Solution Steps

To determine if a function has an inverse, we can use the horizontal line test. If any horizontal line intersects the graph of the function at most once, then the function has an inverse. In other words, a function has an inverse if it is a one-to-one function.

The graph of $g(x) = x^3$ is shown. It can be visually inspected that any horizontal line will intersect the graph at only one point. For example, the line $y=0$ intersects the graph at $x=0$. The line $y=8$ intersects the graph at $x=2$. The line $y=-8$ intersects the graph at $x=-2$. In general, the line $y=c$ will intersect the graph at $x = \sqrt[3]{c}$.

Thus, $g(x) = x^3$ has an inverse. The inverse is $g^{-1}(x) = \sqrt[3]{x}$.

3. Final Answer

Yes, this function has an inverse.