Transcript 5 3 Inverse Functions

5-3: Inverse Functions
Objectives:
Assignment:
1. To find and verify inverse
functions
• P. 347-349: 8, 9-12, 13,
15, 23, 25, 31, 35, 49, 51,
63, 93, 101-104, 109
2. To find the derivative of
an inverse function
• P. 349: 71, 73, 75, 76, 79,
81, 95, 107
Warm Up
Let 𝑦 = 𝑚𝑥 + 𝑏. Find the inverse of 𝑦. What
is the relationship between the slopes of
inverse linear functions?
You will be able to
find and verify
inverse functions
Objective 1
Inverse Relations
An inverse relation is a relation that switches the
inputs and output of another relation.
Inverse
relations
“undo” each
other
Inverse Functions
If a relation and its inverse are both
functions, then they are called inverse
functions.
𝑓 𝑓 −1 𝑥
= 𝑥 and 𝑓 −1 𝑓 𝑥
=𝑥
f -1 = “f inverse” or “inverse of f ”
The inverse of a
function is not
necessarily a function.
Inverse Functions
If a relation and its inverse are both functions,
then they are called inverse functions.
𝑓 𝑓 −1 𝑥
= 𝑥 and 𝑓 −1 𝑓 𝑥
=𝑥
For 𝑓 and 𝑓 −1 to be inverse
functions, the domain of 𝑓
must be equal to the range
of 𝑓 −1 , and the range of 𝑓
must be equal to the
domain of 𝑓 −1 .
If 𝑔 is the
inverse of 𝑓,
then 𝑓 is the
inverse of 𝑔.
Exercise 1
Verify that the functions
𝑓 𝑥 = 2𝑥 3 − 1 and
𝑔 𝑥 =
3
𝑥+1
2
are
inverse functions of
each other.
Graphing Investigation
Suppose we drew a
triangle on the
coordinate plane.
Geometrically
speaking, what
would happen if we
switched the x- and
the y-coordinates?
𝑥, 𝑦 → 𝑦, 𝑥
Graphing Investigation
Suppose we drew a
triangle on the
coordinate plane.
Geometrically
speaking, what
would happen if we
switched the x- and
the y-coordinates?
This
what
𝑥, 𝑦is →
𝑦, happens
𝑥
with inverses.
Graphs of Inverse Functions
Therefore, the graphs
of inverse functions
are reflections across
the line 𝑦 = 𝑥.
Furthermore, to graph
𝑓 −1 , first graph 𝑓 and
then switch the 𝑥- and
𝑦- coordinates of
some key points.
Reflective Property
The graph of 𝑓
contains the point
𝑎, 𝑏 if and only if
the graph of 𝑓 −1
contains the point
𝑏, 𝑎 .
Does it Function?
Recall that we can use the vertical line test to
see if a graph represents a function.
The question
is: How can
we tell if a
function’s
inverse will
be a
function?
Function
Not a Function
Horizontal Line Test
The inverse of a function f is also a function iff no
horizontal line intersects the graph of f more than once.
Horizontal Line Test
The inverse of a function f is also a function iff no
horizontal line intersects the graph of f more than once.
Such a function is
called one-to-one
Every input
has one
output
Every output
has one
input
Existence of an Inverse Function
A function has
an inverse
function if and
only if it is oneto-one.
Always increasing
or decreasing
If 𝑓 is strictly monotonic
on its entire domain,
then it is one-to-one
and therefore has an
inverse function.
Finding the Inverse of a Function
Since the inverse of
a function
switches the 𝑥and 𝑦-values of
the original
function, we can
easily find the
inverse of a
function
algebraically:
Let 𝑓(𝑥) =
Step
𝑦, if 1
necessary
Exchange
Step
2𝑦
the
𝑥 and
variables
Range of 𝑓(𝑥)
Solve
Stepfor
3𝑦
Check
Domain
Exercise 2
Find the inverse of
𝑓 𝑥 = 2𝑥 − 3.
Exercise 3
Use the graph of 𝑦 = sin 𝑥 to explain why its
inverse is not a function.
Exercise 3
What could we do to the graph below so that
its inverse is cleverly a function?
We can restrict
the domain of
the function to an
interval that
would pass the
horizontal line
test.
Master of My Domain
In order for the inverse of 𝑦 = sin 𝑥 to be a function,
we have to restrict the domain. Refer again to the
graph of 𝑓(𝑥) = sin 𝑥 and find a sensible interval for
which 𝑓 −1 is a function.
Objective 2
You will be able to find the
derivative of a function
Continuity and Differentiability
Let 𝑓 be a function whose domain is an interval 𝐼.
If 𝑓 has an inverse function, then the following
statements are true.
If 𝑓 is continuous its domain, then
𝑓 −1 is continuous on its domain.
If 𝑓 is increasing on its domain,
then 𝑓 −1 is increasing on its
domain.
If 𝑓 is decreasing on its domain,
then 𝑓 −1 is decreasing on its
domain.
If 𝑓 is differentiable on an interval
containing c and 𝑓′(𝑐) ≠ 0, then
𝑓 −1 is differentiable at 𝑓(𝑐).
Investigation
Let’s investigate the derivatives of inverse function.
Find the derivative of each function and then
complete the tables below.
𝑓 𝑥 = 𝑥3
𝒙
𝟎
𝟏
𝑔 𝑥 = 𝑥 1/3
𝟐
𝟑
𝒙
𝑓(𝑥)
𝑔(𝑥)
𝑓′(𝑥)
𝑔′(𝑥)
𝟎
𝟏
𝟖
𝟐𝟕
Derivatives of Inverse Functions
Let 𝑓 be a function
that is differentiable
on an interval 𝐼. If 𝑓
has an inverse
function 𝑔, then 𝑔 is
differentiable at any
𝑥 for which
𝑓′ 𝑔 𝑥 ≠ 0, and
𝑔′ 𝑥 =
1
𝑓′ 𝑔 𝑥
𝑓′ 𝑔 𝑥
≠0
Derivatives of Inverse Functions
Let 𝑓 be a function
that is differentiable
on an interval 𝐼. If 𝑓
has an inverse
function 𝑓 −1 , then
𝑓 −1 is differentiable
at any 𝑥 for which
𝑓′ 𝑓 −1 𝑥 ≠ 0, and
𝑓
−1
′ 𝑥 =
1
𝑓′ 𝑓 −1 𝑥
𝑓′ 𝑓 −1 𝑥
≠0
Exercise 4
Let 𝑓 𝑥 =
1 3
𝑥
4
+ 𝑥 − 1.
1. What is the value of 𝑓 −1 𝑥 when 𝑥 = 3?
2. What is the value of 𝑓 −1 ′ 𝑥 when 𝑥 = 3?
Exercise 5: AP
Let 𝑓 be a differentiable
function such that
𝑓 3 = 15, 𝑓 6 = 3,
𝑓′ 3 = −8, and
𝑓′ 6 = −2. The
function 𝑔 is
differentiable and
𝑔 𝑥 = 𝑓 −1 𝑥 for all 𝑥.
What is the value of
𝑔′ 3 ?
A.
B.
C.
D.
1
−
2
1
−
8
1
6
1
3
E. The value of 𝑔′ 3
cannot be determined
from the information
given.
5-3: Inverse Functions
Objectives:
Assignment:
1. To find and verify inverse
functions
• P. 347-349: 8, 9-12, 13,
15, 23, 25, 31, 35, 49, 51,
63, 93, 101-104, 109
2. To find the derivative of
an inverse function
• P. 349: 71, 73, 75, 76, 79,
81, 95, 107