Creating the graph of an inverse function involves swapping the $x$ and $y$ coordinates of points from the graph of the original function. Here's how to do it:

Method

To graph the inverse $y=f^{-1}(x)$  of a function $f(x)$, you do the following

1. Find the inverse function
2. Create a table of values: Choose input values ($x$) for the original function and compute their corresponding output values ($y$). Then switch the roles of $x$ and $y$ to get the points for the inverse function.
3. Plot and connect the points: Use the points to determine the shape of the graph, which you can then draw in the coordinate plane.

Note

The graph of the inverse function is symmetric to the original function over the line $y=x$.

This means if you have the graph of the original function $y=f(x)$, you can find the graph of the inverse function by reflecting over the line $y=x$.

Example

Step 1: Original function and its inverse

We use $y=2x-1$ and its inverse $f^{-1}(x)=\frac{x+1}{2}$.

Step 2: Table of values

Create a table of values for the original function $y=2x-1$:

Now switch $x$ and $y$ to get the table for the inverse function $f^{-1}(x)=\frac{x+1}{2}$:

Step 3: Plot the points

Plot the points $(x,y)$ for $f^{-1}(x)=\frac{x+1}{2}$: $(-1,0)$, $(1,1)$, $(3,2)$, $(5,3)$.

Here the original function $f(x)=2x-1$ has also been plotted. You can see that the inverse function is a reflection of the original function over the line $y=x$.