# Definition of Inverse Functions

$$ \Large+ \leftarrow \text{ inverse operation } \rightarrow -$$

$$ \Large \times \leftarrow \text{ inverse operation } \rightarrow \div $$

$$ \Large x^2 \leftarrow \text{ inverse operation } \rightarrow \sqrt{x} $$

The function $y=4x-1$ can be undone by its inverse function $y=\dfrac{x+1}{4}$.

We can consider this act as two processes or machines. If the machines are inverses then the second machine undoes what the first machine does. No matter what value of $x$ enters the first machine, it is returned as the output from the second machine.

If $(x,y)$ lies on $f$, then $(y,x)$ lines on $f^{-1}$. Reflecting the function in the line $y=x$ has the algebraic effect of interchanging $x$ and $y$.

For instance, $f:y=4x-1$ becomes $f^{-1}:x=4y-1$.

$$\text{The domain of } f^{-1} = \text{ the range of }f$$

$$\text{The range of } f^{-1} = \text{ the domain of }f$$

$y=f^{-1}(x)$ is the inverse of $y=f(x)$ as:

- it is also a function
- it is the relfection of $y=f(x)$ in the line $y=x$

The parabola shown in red below is the reflection of $y=f(x)$ in $y=x$, but it is not the inverse function of $y=f(x)$ as it fails the vertical line test. In this case, the function $y=f(x)$ does not have an inverse.

Now consider the same function $y=f(x)$ but with the restricted domain $x \ge 1$.

The function does now have an inverse function, as show below.

The reciprocal funciton $f(x)=\dfrac{1}{x},x \ne 0$, is said to be a self-inverse function as $f=f^{-1}$.

This is because the graph of $y=\dfrac{1}{x}$ is symmetrical about the line $y=x$.

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume