Inverse Functions – Ultimate Guide of Definition and Graphs

Inverse Functions
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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 $f^{-1}:x=4y-1$.
$$ \large \begin{align} \text{The domain of } f^{-1} &= \text{ the range of }f \\
\text{The range of } f^{-1} &= \text{ the domain of }f \end{align} $$

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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 $y=x$.

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