Lesson #3B: Inverse Functions
Lesson #3B:
Inverse Functions
Day 1
What is the inverse of a function?
A function and its inverse can be described as
the “DO” and the “UNDO” functions.
Remember: A function takes a starting value, performs an
operation, and creates an output answer.
However, the inverse of a function takes the output
value, performs an operation, and arrives at the
original starting value.
Algebra II with Trig: Unit 1
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Lesson #3B: Inverse Functions
What is the inverse of a function?
If functions,
and
, are inverse functions, then
*x is called the identity element*
Think of them as “un-doing” one another and leaving you right
where you started.
Recall: Additive inverses:
Multiplicative inverses:
0 and 1 are the identity elements!
Inverse Notation:
Given
is a function. The inverse of
is represented as
.
The Inverse of a Function
In order to find the inverse of a relation or a function,
swap or switch
and .
If the relation is an equation, solve for .
Example 1: Given the relation {(1, 2), (3, 2), (3, 1)}.
Find the inverse.
Switch
and !
{(2, 1), (2, 3), (1, 3)}
Algebra II with Trig: Unit 1
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Lesson #3B: Inverse Functions
Note: The inverse will be
a relation, but may not
necessarily be a function.
The Inverse of a Function
Example 2: Given the relation,
Find the inverse.
1. Remember:
2. Switch
and
.
.
to find the inverse.
3. Solve for .
Algebra II with Trig: Unit 1
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Lesson #3B: Inverse Functions
The Inverse of a Function
.
Example 3: Given the relation,
. Recall:
Find
1. Remember:
2. Switch
and
inverse function notation
.
to find the inverse.
3. Solve for .
The Inverse of a Function
Example 4: Given the relation,
.
Find
Switch
.
and
to find the inverse.
Solve for .
Algebra II with Trig: Unit 1
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Lesson #3B: Inverse Functions
The Inverse of a Function
Example 5: Find
Recall:
∘
.
∘
Definition of a
function and its
inverse!
∘
The Inverse of a Function
Example 6: Find
Recall:
∘
∘
Algebra II with Trig: Unit 1
∘
.
Definition of a
function and its
inverse!
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Lesson #3B: Inverse Functions
The Inverse of a Function
Example 7: Given:
Show that
and
are inverse functions of each
other by using composition of a function and its inverse.
Recall:
8
3
3
8
3
8
3
8
3
8
8
3
∴

8
8
3
3

and
are
inverse functions.
The Inverse of a Function
Example 8: Given:
Show that
and
are inverse functions of each
other by using composition of a function and its inverse.
Recall:
5
9
9 5
5 9
32
Algebra II with Trig: Unit 1
32
32
32
32

5 9
9 5
∴
9
5
32
32
32

and h
are
inverse functions.
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Lesson #3B: Inverse Functions
Lesson #3B:
Inverse Functions
Day 2
What is a One-to-One Function?
Recall: The inverse of a function may not always be
a function!
The original function must be a one-to-one
function to guarantee that its inverse will
also be a function.
A function is one-to-one if and only if each second
element corresponds to one and only one first element.
(Each and value is used only once and passes the
horizontal line test.)
Algebra II with Trig: Unit 1
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Lesson #3B: Inverse Functions
Inverse of a Function
Example 1) Is the inverse of this
function guaranteed to be a function?
Explain.
No, because is not one-to-one
(does not pass horizontal line test).
Example 2) Is the inverse of this
function guaranteed to be a function?
Explain.
Yes, because is one-to-one
(passes horizontal line test).
The Inverse of a Function
1with
Example 3: Given the function
*What does
. Recall:
(a) Evaluate
1. Remember:
2. Switch
and
.
look like?
inverse function notation
*Why is this
domain given?
*
is the right
side of a parabola
to find the inverse.
[including a vertical
translation down].
3. Solve for .
*However, think about
domain restriction of
*If we take square root of
, we must introduce .
Algebra II with Trig: Unit 1
0.
is now
one-to-one!
[
is
guaranteed to
be a function].
*
!
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Lesson #3B: Inverse Functions
The Inverse of a Function
1with
Example 3: Given the function
∘
(b) Evaluate
.
0.
∘
(c) Evaluate
.


The Inverse of a Function
1with
Example 3: Given the function
0.
(d) What do your results from part (b) and (c) mean?
∘
Algebra II with Trig: Unit 1
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Lesson #3B: Inverse Functions
The Inverse of a Function
.
Example 4: Given the function
(a) What is the domain and range of
Domain:
?
4, ∞
Range: 0, ∞
(b) Sketch the graph of
.
The Inverse of a Function
.
Example 4: Given the function
.
(c) Find
Inverse:
4
Solve for : Square both sides
4
4
4
4
?
(d) What is the domain and range of
Switch
and
to find the inverse → Switch domain and range!
Domain: 0, ∞
(e) Sketch the graph of
axes as
.
Algebra II with Trig: Unit 1
Range:
4, ∞
on the same set of
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Lesson #3B: Inverse Functions
The Inverse of a Function
Example 4: Given the function
(e) Sketch the graph of
axes as
.
.
on the same set of
Inverse Functions Graphically
If a function has an inverse function, the reflection
of that original function in the identity line,
,
will also be a function. (Switch the coordinates of and )
Example:
The red dashed line will not
pass the vertical line test for
functions, thus y = x2 does
not have an inverse function.
You can see that the inverse
exists, but it is NOT a
function.
Algebra II with Trig: Unit 1
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Lesson #3B: Inverse Functions
Inverse Functions Graphically
Example : Are the following graphs inverses of each other?
(a)
(b)
Yes, they are reflections
over the line y = x.
Algebra II with Trig: Unit 1
No, they are no reflections
over the line y = x.
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