Inverse Functions
First and Last Names _ _ _ _ _ __
Learning Outcomes included:
• Simplifying Expressions
• Mathematical Communication
• Basics of Graphing
• Domains
• Problem Structure
• Inverse Functions
Instructions. In this activity, work through each problem with your partner.
Take turns authoring the work. If you have questions, raise your hand. There
are 15 pages to this assignment. Your grade on this assignment is based on
completion.
Model 1: One-to-one-Functions
Recall that a relation is considered a function when each input has exactly one output.
Graphically a function passes the vertical line test.
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One-to-One
Functions:
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~ II '- ,;i
Not
One-to-One
Functions:
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(Pro blems 1-5 will refer to Model 1)
1. Notice that horizontal lines are drawn on the first graph under 'One-to-One Functions'
and on the first graph under 'Not One-to-One Functions'.
(a) How many times does each horizontal line touch the blue graph in the 'One-to-One
Functions• graph?
(b) How many times does each horizontal line touch the blue graph in the 'Not Oneto-One Functions' graph?
2. Draw horizontal lines - anywhere you choose - on each of the remaining graphs in
Model 1 to perform the Horizontal Line Test. What do you notice about the graphs
that are one-to-one functions versus those that are not one-to-one functions?
lnvcrsc Functions
Fall 2026
Model 3: Finding Inverse Functions
H a function is one-to-one then its inverse is also a function
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Example 1: J(x) =2x - 4 Example 2! g(x) =
Steps
Replace f (x) with y.
y=2x-4
2x
Y- -- X + 1
Interd1ange x and y.
X = 2y- 4
x=
2x
x+l
2y
y+l
x(y+1)=2y
Solve for y.
X + 4 = 2y
xy + X = 2y
1
-x + 2 = y
xy-2y = -x
2 1
y = x+2
y(x - 2) = -x
2
-x
Y - x-2
1
f- 1(x) = -X
+2
2
Replace y ,.vith 1-1 (x) 1
o-l(x)
0
= -x
x-2
I
Graphs of g(x) and g- 1 (x)
Graphs off (x) and 1- 1(x)
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5
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10
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-10
(Problems 10-14 will refer to Model 3)
10. In Example 1 of Model 3, where does the equation x = 2y - 4 come from?
11. How is an inverse denoted in function notation? _ _ _ __
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