# How to Graph Inverse Functions

The inverse of a function "undoes" the operation of the original function. If the original function takes an input and produces an output, the inverse function takes that output and produces the original input. Here's a step-by-step guide to graphing the inverse function:

## 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:

- Pick a point \((a,b)\) on the original function’s graph.
- 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.

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