Inverse Functions – Ultimate Guide of Definition and Graphs

Inverse Functions
YouTube player

Domain and Range of Inverse Functions

$$ \Large \begin{align} + \leftarrow &\text{ inverse operation } \rightarrow – \\
\times \leftarrow &\text{ inverse operation } \rightarrow \div \\
x^2 \leftarrow &\text{ inverse operation } \rightarrow \sqrt{x} \end{align} $$
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, 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 $f^{-1}:x=4y-1$.
$$ \large \begin{align} \text{The domain of } f^{-1} &= \text{ the range of }f \\
\text{The range of } f^{-1} &= \text{ the domain of }f \end{align} $$

YouTube player

Graphs of Inverse Functions

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

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

The parabola shown in red below reflects $y=f(x)$ 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 shown 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 $y=x$.

YouTube player


Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function 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 Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

Related Articles


Your email address will not be published. Required fields are marked *