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.

Step-by-step 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,

$\pm \sqrt{x}=y$

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,

$x^{\frac{1}{3}}=y$

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.