Question

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

Answer 1

Answer:

Because its graph will not pass the vertical line test.

Explanation:

We must first find the inverse of the given functions.

Let


`y=f(x)`.

Then for the first function, we have
`y=x^2`

We interchange x and y to get,


`x=y^2`

We make y the subject to get,


`\pm √(x)=y`

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


`y=g(x)`

Then,


`y=x^3`

We interchange x and y to get,


`x=y^3`

We make y the subject to get,


`x^{(1)/(3)}=y`

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.