Unit 3 – Transformations Inverse Functions (Unit 3.3)

Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Lesson Goals
Unit 3 – Transformations
Inverse Functions (Unit 3.3)
When you have completed this lesson you will:
William (Bill) Finch
Mathematics Department
Denton High School
I
Find inverse functions informally, graphically, and
algebraically.
I
Identify when a function is one-to-one with the Horizontal
Line Test.
I
Determine if two functions are inverses of each other.
W. Finch
DHS Math Dept
Inverses
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Inverse Functions
f
inverse).
Inverses
Inverse Graphs
One-to-One
Inverse Alg
Summary
Let f and g be two functions such that
f −1 (x) = x + 3
Domain
Range
Domain
Range
4
1
1
4
5
2
2
5
6
3
3
6
7
4
4
7
f (g (x)) = x
for every x in the domain of g
g (f (x)) = x
for every x in the domain of f
and
the functions f and g are each the inverse function of each
other.
Notice function f takes an input and then subtracts 3, while
f −1 “undoes” f by using the output of f as it’s input and
adding 3.
W. Finch
2 / 16
Inverse
Definition of Inverse Function
Two functions are shown below: f and f −1 (read
f (x) = x − 3
Introduction
Thus you may say either that
g (x) = f −1 (x)
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3 / 16
W. Finch
Inverses
or
f (x) = g −1 (x)
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Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Example 1
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Example 2
Informally determine the inverse of f (x) = 2x − 1.
Show that functions f and g are inverse functions.
√
f (x) = x 3 + 1
g (x) = 3 x − 1
W. Finch
Then verify that both f (f −1 (x)) and f −1 (f (x)) are both equal
to the identity function.
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Inverses
Introduction
Introduction
5 / 16
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
The Graph of an Inverse Function
W. Finch
DHS Math Dept
Inverses
Introduction
6 / 16
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Example 3
The graphs of inverse functions are reflections wrt to the line
y =x.
Sketch the graphs of the inverse functions on the same
coordinate system and show that the graphs are reflections
around the line y = x.
√
f (x) = x 2 , x ≥ 0
f −1 (x) = x
y
y = f (x)
y =x
(a, b)
(b, a)
y = f −1 (x)
x
W. Finch
Inverses
DHS Math Dept
7 / 16
W. Finch
Inverses
DHS Math Dept
8 / 16
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Horizontal Line Test
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
One-to-One Function
A function f has an inverse function if and only if a horizontal
line intersects the graph of f at no more than one point.
A function f is one-to-one if each value of x corresponds to
one value of y , and if each value of y corresponds with one
value of x.
y
y
y
y
y = f (x)
y = f (x)
y = f (x)
y = f (x)
(a, b)
(a, b)
(c, b)
x
(a, b)
(a, b)
(c, b)
x
x
x
f has an inverse function.
f does not have an inverse
function.
W. Finch
f is one-to-one.
DHS Math Dept
Inverses
9 / 16
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Example 4
f is not one-to-one.
W. Finch
DHS Math Dept
Inverses
10 / 16
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Example 5
Determine whether the function has an inverse function (in
other words, determine if the function is one-to-one).
Determine whether the function has an inverse function (in
other words, determine if the function is one-to-one).
y
x
g (x)
10
8
−3 −2 −1 0 1 2 3
9
4
1 0 1 4 9
6
4
2
f (x) = x 2 , x ≥ 0
x
2
W. Finch
Inverses
4
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11 / 16
W. Finch
Inverses
DHS Math Dept
12 / 16
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Finding an Inverse Function Algebraically
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Example 6
Starting with the equation of a function f :
Find the inverse of f (x) =
2x − 5
.
3
1. Use the Horizontal Line Test to determine if f has an
inverse function.
2. Replace f (x) with y .
3. Interchange the roles of x and y , and solve for y .
4. Replace y with f −1 (x) .
5. Verify that f (f −1 (x)) = f −1 (fx)) = x .
W. Finch
DHS Math Dept
Inverses
Introduction
13 / 16
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
Example 7
Inverses
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Inverses
14 / 16
Introduction
Inverse
Inverse Graphs
One-to-One
Inverse Alg
Summary
What You Learned
You can now:
Find the inverse of g (x) = (x − 3)2 , x ≥ 3 .
W. Finch
W. Finch
DHS Math Dept
15 / 16
W. Finch
Inverses
I
Find inverse functions informally, graphically, and
algebraically.
I
Identify when a function is one-to-one with the Horizontal
Line Test.
I
Determine if two functions are inverses of each other.
I
Do problems Chap 1.7 #1, 7, 15-23 odd, 27, 31, 35, 51,
53, 59, 69, 71
DHS Math Dept
16 / 16

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