Transcript 6.4: Inverse Functions 6.5: Graph Square/Cube Root Functions

Inverse Functions
Graph Square/Cube Root Functions
Objectives:
1. To find the inverse of a function
2. To graph inverse functions
3. To graph square and cube root functions as
transformations on parent functions
You will be able to
find the inverse of
a function
Objective 1
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 1
Let f(x) = 4x + 2 and g(x) = 1/4x – 1/2. 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
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 .
Exercise 2
Verify that f(x) = 2x + 3 and f -1(x) = ½x – 3/2
are inverse functions.
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, if 1
Step
necessary
Exchange
the 𝑥 and
Step 2
𝑦
variables
Solve
Stepfor
1𝑦
Exercise 3
Find the inverse of f(x) = −(2/3) x + 2.
Exercise 4
Let 𝑦 = 𝑚𝑥 + 𝑏. Find the inverse of 𝑦. What
is the relationship between the slopes of
inverse linear functions?
Exercise 5
Find the inverse of the given function.
1. 𝑓 𝑥 = 𝑥 + 4
2. 𝑓 𝑥 = 2𝑥 − 1
3
3. 𝑓 𝑥 = − 2 𝑥 + 1
Inverses of Nonlinear Functions
For some nonlinear functions, you have to
first restrict the domain to find its inverse.
𝑦 = 𝑥2
Domain: ℝ
𝑥 = 𝑦2
± 𝑥=𝑦
𝑦 = 𝑥 or 𝑦 = − 𝑥
?
Inverses of Nonlinear Functions
For some nonlinear functions, you have to
first restrict the domain to find its inverse.
𝑦 = 𝑥2
Domain: 𝑥 ≥ 0
𝑥 = 𝑦2
± 𝑥=𝑦
𝑦 = 𝑥 or 𝑦 = − 𝑥
Inverses of Nonlinear Functions
For some nonlinear functions, you have to
first restrict the domain to find its inverse.
𝑦 = 𝑥2
Domain: 𝑥 ≤ 0
𝑥 = 𝑦2
± 𝑥=𝑦
𝑦 = 𝑥 or 𝑦 = − 𝑥
Exercise 6
Find the inverse of the given function.
1. 𝑓 𝑥 = 𝑥 6 , 𝑥 ≥ 0
1
2. 𝑓 𝑥 = 27 𝑥 3
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 is what happens
𝑥,with
𝑦 inverses.
→ 𝑦, 𝑥
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 𝑥and 𝑦- coordinates of
some key points.
Exercise 7
The graph shows
𝑓(𝑥). Graph the
inverse of 𝑓(𝑥).
Is the inverse a
function?
Exercise 8
Let f(x) = ½ x – 5.
1. Find f -1
2. State the
domain of each
function
3. Graph f and f -1
on the same
coordinate
plane
Exercise 9a
1.
2.
3.
For an input of 2,
what is the output? Is
it unique?
For an output of 8,
what was the input?
Is it unique?
What does the
answer to Q2 tell you
about the inverse of
the function?
Exercise 9b
Let f(x) = x3.
1. Find f -1
2. State the
domain of each
function
3. Graph f and f -1
on the same
coordinate
plane
Exercise 10a
1.
2.
3.
For an input of 4,
what is the output? Is
it unique?
For an output of 16,
what was the input?
Is it unique?
What does the
answer to Q2 tell you
about the inverse of
the function?
Exercise 10b
Let f(x) = x2.
1. Find f -1
2. State the
domain of each
function
3. Graph f and f -1
on the same
coordinate
plane
Does it Function?
As the previous Exercise demonstrated,
even though you can find the inverse of a
function, the inverse itself may not be a
function.
Remember, we overcome this shortcoming
by restricting the domain of the original
function.
Inverses of Nonlinear Functions
For some nonlinear functions, you have to first
restrict the domain to find its inverse.
𝑦 = 𝑥2
𝑥 = 𝑦2
Domain: ℝ
± 𝑥=𝑦
𝑦 = 𝑥 or 𝑦 = − 𝑥
?
Inverses of Nonlinear Functions
For some nonlinear functions, you have to first
restrict the domain to find its inverse.
𝑦 = 𝑥2
𝑥 = 𝑦2
Domain: 𝑥 ≥ 0
± 𝑥=𝑦
𝑦 = 𝑥 or 𝑦 = − 𝑥
Inverses of Nonlinear Functions
For some nonlinear functions, you have to first
restrict the domain to find its inverse.
𝑦 = 𝑥2
𝑥 = 𝑦2
Domain: 𝑥 ≤ 0
± 𝑥=𝑦
𝑦 = 𝑥 or 𝑦 = − 𝑥
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.
Exercise 11
Graph the function f. Then use the graph to
determine whether f -1 is a function.
1
1. 𝑓 𝑥 = 5 𝑥 5
2. 𝑓 𝑥 = 𝑥 + 4
3. 𝑓 𝑥 = 2𝑥 6 , 𝑥 ≥ 0
One-to-One Function
If f passes both the
vertical and the
horizontal line tests—
that is, if f and f -1 are
functions—then f is a
one-to-one function.
Every input has
exactly one output
Every output has
exactly one input
Radical Parents
Square root parent
function: 𝑓 𝑥 = 𝑥
Cube root parent
function: 𝑓 𝑥 =
3
𝑥
We can perform transformations on these parent functions
to help graph whole families of radical functions.
Radical Parents
Use the following to
discover the roll of a,
h, and k in the
following radical
functions:
𝑦 =𝑎 𝑥−ℎ+𝑘
Scaling
a
• 0 < |a| < 1: Shrink vertically
• |a| > 1: Stretch vertically
• a < 0: Flips
h
Horizontal translation
k
Vertical translation
3
𝑦 =𝑎 𝑥−ℎ+𝑘
Exercise 12
Graph the following
radical function.
Then state the
domain and range.
𝑦 =3 𝑥+2−4
Exercise 13
Graph the following radical function. Then state the
domain and range.
13
𝑦=−
𝑥−2+3
2
Exercise 14
Graph the following radical functions. Then state the
domain and range.
1.
2.
𝑦 = −4 𝑥 − 1 + 2
3
𝑦 =2 𝑥+4−3
Inverse Functions
Graph Square/Cube Root Functions
Objectives:
1. To find the inverse of
a function
2. To graph inverse
functions
3. To graph square and
cube root functions as
transformations on
parent functions