Explain why the inverse of  f(x)=x^2 is not a function, but the inverse of  g(x)=x^3 is a function.  

Because its graph will not pass the vertical line test.

Step-by-step explanation:

We must first find the inverse of the given functions.

Let

.

Then for the first function, we have

We interchange x and y to get,



We make y the subject to get,



This is not a function because one x-value is mapping onto two y-values.

Hence its graph will not pass the vertical line test.
See red graph.

For the second function, we again let



Then,



We interchange x and y to get,



We make y the subject to get,



This is a function because, one x-value maps on to one and only one y-value. This tells us that the graph of this function will past the vertical line test.

see blue graph.