How to Graph Inverse Functions

Step-by-step Guide to Graph Inverse Functions

Here is a step-by-step guide to graph inverse functions:

Step 1: Understand the Inverse Function

Given a function \(f\), its inverse is often denoted as \(f^{−1}\). The inverse function essentially swaps the x-values with the y-values of the original function. So, if \(f(a)=b\), then \(f^{−1}(b)=a\).

Step 2: Check for Invertibility

Before finding the inverse, ensure that the function is invertible. A necessary condition for a function to have an inverse is that it must be bijective, i.e., both injective (one-to-one) and surjective (onto). Graphically, a function is one-to-one if and only if it passes the horizontal line test.

Step 3: Swap x and y

For the given function, interchange \(x\) and \(y\). If you’re given \(y=f(x)\), swap to get \(x=f^{−1}(y)\).

Step 4: Solve for the Inverse

Once you’ve interchanged \(x\) and \(y\), solve for \(y\) to express \(y\) in terms of \(x\). This new expression represents the inverse function.

Step 5: Graph the Original and Inverse Function

Graph both the original function and its inverse on the same set of axes.

Step 6: Recognize the Reflection

The graph of the inverse function is a reflection of the graph of the original function over the line \(y=x\). This is because the \(x\) and \(y\) values are swapped in the inverse function.

Step 7: Test the Inverse Graphically

To verify the accuracy of the inverse:

1. Pick a point \((a,b)\) on the original function’s graph.
2. Check that the point \((b,a)\) exists on the graph of the inverse function.

Step 8: Final Observations

When observing the graphs:

• If the original function is increasing, the inverse function will be increasing as well.
• If the original function is decreasing, the inverse function will be decreasing.
• If the original function has horizontal asymptotes, the inverse will have vertical asymptotes, and vice versa.