Inverse Functions Revision Notes for Leaving Cert Mathematics

Inverse Functions

The inverse of a function reverses the original function's input-output relationship. If a function $f(x)$ maps $x \to y$ , its inverse $f^{-1}(x)$ maps $y \to x$ .

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Finding the inverse of a function

Replace $f(x)$ with $y$
Swap $x$ and $y$
Solve for $y$
Replace $y$ with $f^{-1}(x)$

Example

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If $f(x)=2x+3$ , find $f^{-1}(x)$ .

First, replace $f(x)$ with $y$ :

y=2x+3

Swap $x$ and $y$ :

x=2y+3

Solve for $y$ :

x=2y+3 \implies y=\frac{x-3}{2}

Write the inverse :

f^{-1}(x)=\frac{x-3}{2}

Now I can find the original input to any output. For example :

f(1)=2(1)+3=5

f^{-1}(5)=\frac{(5)-3}{2}=1

$1$ maps to $5$ , $5$ maps back to $1$ .

A function only has an inverse if it is one-to-one ( $1$ -to- $1$ ). For example, $f(x) = x^2, x \in \mathbb{R}$ has no inverse. This function is many-to-one, therefore no inverse exists.

Geometric Relationship between a Function and its Inverse

To find the inverse of a function, we simply swap its $x$ and y values. This is the equivalent of reflecting the graph through the line $y = x$ .

Inverse Functions (Leaving Cert Mathematics): Revision Notes