Problem

Find the inverse of

and find the domains of f and f ^–1.

Solution

To find the inverse of f, set y equal to f(x), then exchange x and y and solve for y.

This is the inverse function of f.

Since f(x) is a fraction, and the denominator can never equal 0, its domain is all real numbers. f ^–1, however, contains a logarithm, so we must make sure that we are never taking a logarithm of 0 or a negative number. We require

However, x is already positive (go back to the first line in the solution where we interchanged x and y). This inequality is solved for all x, so the domain of f ^–1(x) is all real numbers.