Question

What is the inverse function of `f(x)=3x^4`?

Answer 1

With its natural domain of `(-oo, oo)`, that function is not one-to-one. It does not have an inverse.

Explanation:

If we restrict the domain to some subset of `[0,oo)`, then the function has an inverse:

`y = 3x^4`

`x^4 = y/3`,

`x = root(4)(y/3)`

So the inverse function is `g(x) = root(4)(x/3)`
(Rationalize the denominator if you wish to get `g(x) = root(4)(27x)/3`.)

On the other hand:
It we restrict the domain to a subset of `(-oo, 0]`, then the inverse is `h(x) = -root(4)(x/3)`