Pre-Calculus – Inverse Functions
Name________________________
This activity is designed to help you gain an understanding of inverse functions by first observing the inverse
of a function graphically, and then calculating the inverse algebraically.
A. Your group’s functions are (each of you should list them in the same order):
1. _________________________
2. _________________________
3. _________________________
B. Open a new file in Geogebra. Save this file as “Function 1.”
1. Using the Input Bar at the bottom of the screen, graph function #1 as
f (x) = __________________.
2. Plot a point “A” on this function. Make sure that f (x) highlights when you plot A, so that A is

literally a point on f (x) .
3. Double click on A to open “Object Properties.” Under the “Style” tab, change the size of A to 7
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and the color (YOU can pick the color!) to make it more visible.

4. Using the Input Bar at the bottom of the screen, plot a new point B = (y(A), x(A)). This
point should not look like it is “attached” to anything.
5. Double click on B to open “Object Properties.”
i. Under the “Basic” tab, click the box “Show Trace.”
ii. Change the size of B to 5 and choose a new color to also make it more visible.
6. Make sure the top left icon in the toolbar (that looks like an arrow) is highlighted. Click and
drag on A.
7. In the space below, draw a sketch of your original graph and the graph traced by the point B
on the same axes.
8. Make a table of values for f (x) and the graph that was plotted by tracing point B by referring
to the column on the left side of the Geogebra window. Drag the point A around and list
several of the x- and y-values for A and the corresponding x- and y-values for B (in other

words, when A is a certain point, what are the x- and y-values for point B?).
Function #1 (using point A):
x
Using point B:
x
f (x)

9. How do the points A and B seem to be related?
C. Calculating an Inverse Function
1. Re-write function #1 by “switching” the x and y terms.
Example 1: If you have the function y = 2x, rewrite it as x = 2y.
2. Solve the new equation for y. Then, call it, g(x)
Example 1 (continued): x = 2y  ½(x) = y  g(x) = ½(x)
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
y
D. Graphing the Inverse Function
1. Return to the file saved as “Function 1.” Go to the View menu to “Refresh Views” to erase the
graph traced by the point B. Graph this new equation, g(x) , on the same screen as the original
function #1, f (x) .
i. Make sure the top left box (that looks like an arrow) is highlighted. Click and drag on A.
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ii. What do you notice about g(x) and the graph traced by the point B?
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2. State the domain and range of f (x) , g(x) , and the graph traced by the point B.
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i.
f (x) –
Domain: ________________
Range: ________________
ii.
g(x) –
 ________________
 Domain:
Range: ________________
iii. graph of point B – Domain: ________________
Range: ________________

3. What do you notice about the domain and range of f (x) in relation to the domain and
range of g(x) ?
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
4. Graph the line y = x on the same screen. Describe f (x) and g(x) in relation to this line. Draw
a rough sketch below.
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E. Generalizing Inverse Functions
1. Describe the coordinate pairs of a function [ f (x) ] and its inverse [ g(x) ].
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2. Describe how to find the inverse of a function algebraically.
3. Compare the graphs of a function and its inverse. How are the domain and range of a function
and its inverse related?
F. HW #_______
1. Complete steps A-E for your group’s function #2.
2. Complete steps A-E for y  4 x 2 .
3.
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Functions and their Inverses:
y  3x  2
y
x 2
3

ye

x


y  log x
y  10 x


y
y  23 x
1 3
x
8


2
y  x4
3

3
y  x 6
2

y  ln( 2x)
1
y  ex
2


y

y  ln x
27 3
x
64
y

43
x
3