Inverse of a function

According to the definition of a function, the inverse of a function is also a function if and only if the function is one-to-one.

If the function f is a one-to-one function given by an equation, then its inverse is denoted by $f^{-1}$ which is obtained by interchanging the variables x and y then making y the subject of the formula.

I.e. If $y = f(x)$, then $x = f^{-1}(y)$

Example 1

Find the inverse of each of the following functions:

1. $f(x) = 3x - 6$
2. $f(x) = x^3$

Solution:

Example 2

Find the inverse of the function $f(x) = x - 5$ and then sketch the graph of $f^{-1}(x)$, also state the domain and range of $f^{-1}(x)$.

Solution:

Domain = {All real numbers} Range = {All real numbers}

NB: If a function f takes a domain A to a range B, then the inverse $f^{-1}$ takes B back to A. Hence the domain of $f^{-1}$ is the range of f, and the range of $f^{-1}$ is the domain of f.

Example 3

Solve:

Solutions: