6-1 and 6-2: Inverse Functions

6-1 and 6-2: Inverse Functions
Objectives:
1. To define the inverse
of a function
2. To find the inverse of
a linear function
Assignment:
• P. 97-98: 1-26
• Challenge Problems
You will be able to
define the inverse
of a function
Objective 1
Graphing Investigation
Suppose we drew a
triangle on the
coordinate plane.
Geometrically
speaking, what
would happen if we
switched the 𝑥and the 𝑦coordinates?
𝑥, 𝑦 → 𝑦, 𝑥
Graphing Investigation
Geometrically
speaking, when
you switch the 𝑥and the 𝑦coordinates of a
graph, the graph
will reflect across
the line 𝑦 = 𝑥.
This is what happens
𝑥,with
𝑦 inverses.
→ 𝑦, 𝑥
Graphing Investigation
Geometrically
speaking, when
you switch the 𝑥and the 𝑦coordinates of a
graph, the graph
will reflect across
the line 𝑦 = 𝑥.
This is what happens
𝑥,with
𝑦 inverses.
→ 𝑦, 𝑥
Exercise 1
The graph shows
𝑓(𝑥). Graph the
inverse of 𝑓(𝑥).
Is the inverse 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.
You will be able to
find the inverse of a
linear function
Exercise 2a
1
𝑥
2
Let 𝑓(𝑥) = 2𝑥 − 4 and 𝑔(𝑥) =
+ 2. Find
𝑓(5) and 𝑔(6). Explain the significance of
your answer.
Exercise 2b
1
𝑥
2
Graph 𝑓(𝑥) = 2𝑥 − 4 and 𝑔(𝑥) =
+ 2 on
the same coordinate plane. Describe the
relationship between 𝑓(𝑥) and 𝑔(𝑥).
Function Composition
Function composition happens when we
take a whole function and substitute it in
for x in another function.
ℎ 𝑥 =𝑔 𝑓 𝑥
Substitute 𝑓(𝑥) in for 𝑥 in 𝑔(𝑥)
– The “interior” function gets substituted in for x
in the “exterior” function
Exercise 3
1
Let 𝑓(𝑥) = 2𝑥 − 4 and 𝑔(𝑥) = 2 𝑥 + 2. Find the
following compositions.
1.
𝑓 𝑔(𝑥)
2.
𝑔 𝑓(𝑥)
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
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.
The inverse of a
function is not
necessarily a
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) = 𝑦,
Step
1
if
necessary
Exchange
the
𝑥 and
Step
2
𝑦 variables
Solve
Stepfor
1𝑦
Exercise 4
Find the inverse of 𝑓(𝑥) =
2
−3𝑥
+ 2.
Exercise 5
Verify that 𝑓(𝑥) =
𝑓 −1
𝑥 =
3
− 𝑥
2
2
−3𝑥
+ 2 and
+ 3 are inverses.
Exercise 6
Let 𝑦 = 𝑚𝑥 + 𝑏. Find the inverse of 𝑦. What
is the relationship between the slopes of
inverse linear functions?
6-1 and 6-2: Inverse Functions
Objectives:
1. To define the inverse
of a function
2. To find the inverse of
a linear function
Assignment:
• P. 97-98: 1-26
• Challenge Problems