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# Graphing an Inverse Function

If a function f has an inverse f -1, you can use the graph of f to find the graph of f -1.

Procedure â€” To Graph the Inverse of a Function

Step 1 Identify several points (a, b) on the graph of the given function.

Step 2 Switch the x- and y-coordinates of each point to form the points (b, a) for the inverse function.

Step 3 Plot the new points and connect them with a smooth line.

Note:

Remember, if f contains the ordered pair (a, b), then f -1 contains the ordered pair (b, a).

Example 1

The graph of f(x) = 2x - 5 is shown. Graph f -1. Solution

Step 1 Identify several points (a, b) on the graph of the given function.

From the graph, we choose (0, -5), (1, -3), and (4, 3). Step 2 Switch the x- and y-coordinates of each point to form the points (b, a) for the inverse function. The new ordered pairs are (-5, 0), (-3, 1), and (3, 4).

Step 3 Plot the new points and connect them with a smooth line There is a visual relationship between the graphs of a function and its inverse function. To see this, letâ€™s start with the graphs of f and f -1 from the last example. Look at what happens when we add the graph of y = x. Notice that the graphs of f and f -1 are symmetric about the line y = x.

That is, f -1 is the reflection of f about the line y = x. Similarly, f is the reflection of f -1 about the line y = x.

Note:

If the point (a, b) is on the graph of f, then its mirror image (b, a) is on the graph of f -1.

So, one way to get the graph of f -1 is to start with the graph of f, and reflect it about the line y = x.

Example 2

The graph of f(x) is shown. Graph f -1. Solution

Step 1 Identify several points (a, b) on the graph of the given function.

From the graph, we choose (-2, -6), (-1, 1), (0, 2), (1, 3), and (1.4, 5). Step 2 Switch the x- and y-coordinates of each point to form the points (b, a) for the inverse function.

The new ordered pairs are (-6, -2), (1, -1), (2, 0), (3, 1) and (5, 1.4). Step 3 Plot the new points and connect them with a smooth line. Notice that f and f -1 are symmetric about the line y = x. 