In previous sections, you defined the inverse of a function and found the inverse of the functions represented in tables or from equations. In this section, you will find the inverse of a function that is graphed. Remember, the inverse of a function, f ^-1(x), is the result from exchanging the x- and y-values or they do and undo each other.

On your graph paper, write the equation and then graph it. If you need some graph paper you can print it.

Once you finish graphing, click on the image below and enter the equations one at a time in the box y = on the page.

Note: Type the equations in the form written below.

1. y = (x + 2)/3
2. y = x^2
3. y = (x^3) + 2
4. y = sqrt(x + 5)

After you type the equation into the box, click “Solve” to show the original graph of the function in RED and the inverse is shown in GREEN.

Graph the inverse on your graph paper.

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Conclusion Questions

• Select three coordinates from the first graph and list them. Find the three corresponding coordinates on the inverse of the function.

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  Some sample coordinates are (-2, 0), (1, 1), and (3, 12 over 3 2 3 ). Their corresponding coordinates on the inverse of the function are (0, -2), (1, 1), and (12 over 3 2 3 , 3).

• Describe the relationship between the point on the function and the point on the inverse of the function.

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  The coordinates of the function are the ordered pairs (x, y), and the coordinates of the inverse of the function are the ordered pairs (y, x).

• Plot the points you selected on a separate graph. Determine the line of reflection for the relationship between the function and the inverse of the function.

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  The function and the inverse of the function are reflections across the line y = x.

• How are the graphs of a function and its inverse related?

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  The graph of an inverse of a function f ^-1(x) is the reflection of the graph of the function f(x) across the line y = x.
  
  

Pause and Reflect

Explain how the inverse functions of a table, an equation, and a graph are similar.

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In order to find the inverse of the function, the x- and y-values are exchanged, and the ordered pair (x, y) of the function becomes the ordered pair (y, x) of the inverse of the function.

If you were given a graph and its equation, compare the method for finding the inverse of the function by graphing with the method of finding the inverse of the function by using the equation.

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When graphing, reflecting the function across the line y = x is the same as exchanging the x’s and y’s and the y’s for x’s in the equation. Either method would produce the same inverse of the function.

Practice

1. Graph the inverse of the function y = 1 over 2 1 2 x² – 3.

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   You can use the applet above to find the graph or find the inverse of the equation then graph it.

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   The BLUE graph is the function and the PINK graph is the inverse of the function.
   
   

2. Using the above function, create a table for the function and a table for the inverse function.

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   You can use the applet above to find the graph or find the inverse of the equation then graph it.

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3. Find the inverse of the function of the equation y = 1 over 2 1 2 x² – 3.

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   Exchange the x’s and y’s then solve for y.

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