Question

How do you determine whether the function `f(x)=x^4` has an inverse and if it does, how do you find the inverse function?

Answer 1

With the (implicit) domain `RR`, `f(x)` is not one to one, so its inverse is not a function. If we restrict the domain of `f(x)` then we can define an inverse function.

Explanation:

In order to have an inverse function, a function must be one to one.

In the case of `f(x) = x^4` we find that `f(1) = f(-1) = 1`. So `f(x)` is not one to one on its implicit domain `RR`.

If we restrict the domain of `f(x)` to `[0, oo)` then it does have an inverse function, namely:

`f^(-1)(y) = root(4)(y)`

Some more details...

In general, if you want to find the inverse of a function `f(x)` then set `y = f(x)` and attempt to solve for `x`.

In our example, put:

`y = f(x) = x^4`

Then taking the square root of both ends, allowing for both possible signs we find:

`+-sqrt(y) = x^2`

Assuming that we are only interested in real values of `x`, we require the left hand side to be non-negative, so assume the non-negative square root.

Then take the square root of both sides again to get:

`+-sqrt(sqrt(y)) = x`

That is:

`x = +-sqrt(sqrt(y)) = +-root(4)(y)`

Note that this does not give us a unique value of `x` in terms of `y`.

So the inverse function is not defined, unless we have a restriction such as `x >= 0`, i.e. restrict the domain of `f(x)` to be `[0, oo)`.