We think you are located in South Africa. Is this correct?

Inverse Functions

Test yourself now

High marks in maths are the key to your success and future plans. Test yourself and learn more on Siyavula Practice.

Sign up and test yourself

2.3 Inverse functions (EMCF8)

An inverse function is a function which does the “reverse” of a given function. More formally, if \(f\) is a function with domain \(X\), then \({f}^{-1}\) is its inverse function if and only if \({f}^{-1}\left(f\left(x\right)\right)=x\) for every \(x \in X\).

\begin{align*} & \\ y = f(x) &: \text{indicates a function} \\ & \\ {y}_{1} = f\left({x}_{1}\right) &: \text{indicates we must substitute a specific } x_{1} \text{ value} \\ & \quad \text{into the function to get the corresponding } y_{1}\text{ value} \\ & \\ {f}^{-1}\left(y\right) = x &: \text{indicates the inverse function} \\ & \\ {f}^{-1}\left(y_{1}\right) = {x}_{1} &: \text{indicates we must substitute a specific } y_{1} \text{ value} \\ & \quad \text{into the inverse to return the specific } x_{1}\text{ value} \end{align*}

A function must be a one-to-one relation if its inverse is to be a function. If a function \(f\) has an inverse function \(f^{-1}\), then \(f\) is said to be invertible.

Given the function \(f(x)\), we determine the inverse \(f^{-1}(x)\) by:

  • interchanging \(x\) and \(y\) in the equation;
  • making \(y\) the subject of the equation;
  • expressing the new equation in function notation.

Note: if the inverse is not a function then it cannot be written in function notation. For example, the inverse of \(f(x) = 3x^2\) cannot be written as \(f^{-1}(x) = \pm \sqrt{\frac{1}{3}x}\) as it is not a function. We write the inverse as \(y = \pm \sqrt{\frac{1}{3}x}\) and conclude that \(f\) is not invertible.

If we represent the function \(f\) and the inverse function \({f}^{-1}\) graphically, the two graphs are reflected about the line \(y=x\). Any point on the line \(y = x\) has \(x\)- and \(y\)-coordinates with the same numerical value, for example \((-3;-3)\) and \(\left( \frac{4}{5}; \frac{4}{5} \right)\). Therefore interchanging the \(x\)- and \(y\)-values makes no difference.

88172bd1cada56d990b3b56e1fb57612.png

This diagram shows an exponential function (black graph) and its inverse (blue graph) reflected about the line \(y = x\) (grey line).

Important: for \({f}^{-1}\), the superscript \(-\text{1}\) is not an exponent. It is the notation for indicating the inverse of a function. Do not confuse this with exponents, such as \(\left( \frac{1}{2} \right)^{-1}\) or \(3 + x^{-1}\).

Be careful not to confuse the inverse of a function and the reciprocal of a function:

InverseReciprocal
\(f^{-1}(x)\)\([f(x)]^{-1} = \frac{1}{f(x)}\)
\(f(x)\) and \(f^{-1}(x)\) symmetrical about \(y=x\)\(f(x) \times \frac{1}{f(x)} = 1\)
Example: \(\qquad \qquad \qquad \qquad \qquad \qquad\)Example: \(\qquad \qquad \qquad \qquad \qquad\)
\(g(x) = 5x \therefore g^{-1}(x)= \frac{x}{5}\)\(g(x) = 5x \therefore \frac{1}{g(x)} = \frac{1}{5x}\)