Problem

Let

Find f ^–1(x), and then find the domains of f(x) and f ^–1(x).

Solution

To find the inverse function, let

Then switch x with y, and solve for y.

*

Therefore f ^–1(x) equals

Note that to get equation * we choose the positive square root of 4(x³ + 8). We have to make a choice here, since f(x) is not a one-to-one function. It is symmetric around the line x = –1, so for every number a, f(a) = f(–2 – a).

Well, this causes a problem when computing the inverse, since a rule that produces two outputs is not a function. So we restrict the domain of f(x) to a part of it that is one-to-one, x ³ –1. This translates to restricting the range of the inverse to values greater than or equal to –1, which we get with the positive square root. We could just as easily have chosen the subset of the domain of f(x) with x less than or equal to –1, with the corresponding choice of the negative square root at equation *, giving an alternate definition for the inverse.

Finding the domains

Since we can take the cube root of both positive and negative numbers, the domain of f is all real numbers.

Finding the domain of the inverse requires more care, since it includes a square root. We must have

so

This is the domain of f ^–1(x).