Question

What is the inverse function of `f(x)=x^2`?

Answer 1

The inverse is `f^-1(x)=sqrtx`

Explanation:

Hence `f(x)=x^2=>y=x^2=>sqrty=sqrt(x^2)=>sqrty=absx=>x=+-sqrty`

Answer 2

`f(x) = x^2` is not one-to-one. It does not have an inverse function.

Explanation:

If we try to solve `y=x^2` for `x` we do not get a single value. That means we do not get a function.

We get `x = +- sqrty`.

In order to be invertible a function must be one-to-one.

That means that we must have:
for every `x_1 != x_2`, we get `f(x_1) != f(x_2).`.

For `f(x) = x^2`, we ghave `f(-1) = f(1)` (for example), so there is no inverse function.