Inverse of a Function

The inverse of a function is the function that produces the opposite effect of the original function. That is, every input and its corresponding output for the original becomes an output and its corresponding input for the inverse function. Given a function $f(x)$, the inverse function is represented as $f^{-1}(x)$.

Because of the bijective nature of the original function (every output has exactly one input), the inverse function will also be bijective.

Geometric significance of the inverse

The inverse of a function is the result of reflecting the graph of the function in the line $y=x$. In such a reflection, every point has its $x$ coordinate interchanged with its $y$ coordinate: $$ \begin{equation}\begin{aligned} (a,b)\rightarrow (b,a)\\ \end{aligned}\end{equation} $$

The un-ice-cream-inator

If the ice cream machine is the original function, then the inverse function will be a device that takes the ice cream as input and outputs the corresponding ingredient originally used to produce the ice cream in the first place.

Finding the inverse of a function

In order to find the inverse of a function, use the following steps:

1. Let $f(x)$ be $y$
2. Interchange $x$ and $y$ in the equation (at this point $y$ no longer means $f(x)$ but rather $f^{-1}(x)$)
3. Make $y$ the subject of the formula

Examples

• Find the inverse of the function $f(x)=3x-1$ $$ \begin{equation}\begin{aligned} f(x)&=3x-1\\ \end{aligned}\end{equation} $$ Let $f(x)$ be $y$: $$ \begin{equation}\begin{aligned} y&=3x-1\\ \end{aligned}\end{equation} $$ Interchange $x$ and $y$: $$ \begin{equation}\begin{aligned} x&=3y-1\\ \end{aligned}\end{equation} $$ Transpose for $y$: $$ \begin{equation}\begin{aligned} x+1&=3y\\ y&=\frac{x+1}{3}\\ \therefore f^{-1}(x)&=\frac{x+1}{3}\\ \end{aligned}\end{equation} $$

• Find the inverse of the function $f(x)=\frac{x+1}{5x-4}$ $$ \begin{equation}\begin{aligned} f(x)&=\frac{x+1}{5x-4}\\ \end{aligned}\end{equation} $$ Let $f(x)$ be $y$: $$ \begin{equation}\begin{aligned} y&=\frac{x+1}{5x-4}\\ \end{aligned}\end{equation} $$ Interchange $x$ and $y$: $$ \begin{equation}\begin{aligned} x&=\frac{y+1}{5y-4}\\ \end{aligned}\end{equation} $$ Transpose for $y$: $$ \begin{equation}\begin{aligned} x(5y-4)&=y+1\\ 5xy-4x&=y+1\\ 5xy-y&=1+4x\\ y(5x-1)&=1+4x\\ y&=\frac{1+4x}{5x-1}\\ \therefore f^{-1}(x)&=\frac{4x+1}{5x-1}\\ \end{aligned}\end{equation} $$