MA100,Week 3
Transformations, Operations on Functions, Inverses
Transformations of Functions
Vertical and Horizontal Shifts Suppose
. To obtain the graph of
, shift the graph of
a distance units upward
, shift the graph of
a distance units downward
y
c
0
c
x
1
MA100,Week 3
Vertical and Horizontal Shifts Suppose
y
Transformations, Operations on Functions, Inverses
. To obtain the graph of
, shift the graph of
a distance units to the right
, shift the graph of
a distance units to the left
y
c
0
c
x
2
MA100,Week 3
Example Shifting a Graph
2
Transformations, Operations on Functions, Inverses
Adding 1 to the right-hand side of the formula
2
to get
2
Solution
𝑦 = 𝑥2 + 1
𝑦 = 𝑥2
Shifts the graph up 1 unit
3
MA100,Week 3
Transformations, Operations on Functions, Inverses
2
Example Shifting a Graph
Adding -2 to the right-hand side of the formula
2 to get
2
Solution
𝑦 = 𝑥2
𝑦 = 𝑥2 − 2
Shifts the graph down 2 units
4
MA100,Week 3
Example Shifting a Graph
2 to get
Adding 3 to in
Solution
2
Transformations, Operations on Functions, Inverses
2
Add a positive constant
to 𝑥.
𝑦 = (𝑥 + 3)2
5
MA100,Week 3
Suppose
. To obtain the graph of
, stretch the graph of
, compress the graph of
, compress the graph of
, stretch the graph of
Transformations, Operations on Functions, Inverses
vertically by a factor of
vertically by a factor of
horizontally by a factor of
horizontally by a factor of
𝑦 = 𝑐𝑓(𝑥)
(𝑐 > 1)
𝑦 = 𝑓(𝑥)
1
𝑦 = 𝑓(𝑥)
𝑐
6
MA100,Week 3
Suppose
Transformations, Operations on Functions, Inverses
. To obtain the graph of
, reflect the graph of
about the -axis
, reflect the graph of
about the -axis
7
MA100,Week 3
Example Scaling and Reflecting a Graph
𝑦=3 𝑥
Transformations, Operations on Functions, Inverses
stretches the graph vertically by a factor of 3
𝑦= 𝑥
𝑦=
1
𝑥
3
compresses the graph by a factor of 3
8
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Scaling and Reflecting a Graph
9
MA100,Week 3
Transformations, Operations on Functions, Inverses
𝑦 = 3𝑥
compress the graph
horizontally by a factor
of 3
stretch the graph horizontally
by a factor of 3
𝑦= 𝑥
𝑦
=
𝑥/3
10
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Scaling and Reflecting a Graph
is a reflection
across the -axis
the -axis
is a reflection of
across
11
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Sketch the graph of the function
Solution
We write the equation of the graph as
2
6
2
2
.
2
2
2
2
12
MA100,Week 3
Transformations, Operations on Functions, Inverses
2
3 units to the left
2
𝑦 = 𝑥2 + 6𝑥 + 10
1 unit upward
(- 3, 1)
13
MA100,Week 3
Transformations, Operations on Functions, Inverses
2
Example Sketch the graph of the function
Solution
.
shift to the right 3 units
𝑦 = 𝑥2
𝑦 = (𝑥 − 3)2
reflect in the xaxis and stretch
by a factor of 2
2
2
14
MA100,Week 3
Transformations, Operations on Functions, Inverses
2
shift upward 1 unit
2
15
MA100,Week 3
Theorem
Transformations. Suppose
Transformations, Operations on Functions, Inverses
is a function. If
and
, then to graph
1 Subtract from each of the -coordinates of the points on the graph of .
This results in a horizontal shift to the left if
or right if
.
2 Divide the -coordinates of the points on the graph obtained in Step 1 by .
This results in a horizontal scaling, but may also include a reflection about
the -axis if
.
3 Multiply the -coordinates of the points on the graph obtained in Step 2 by
. This results in a vertical scaling, but may also include a reflection about
the -axis if
.
4 Add to each of the -coordinates of the points on the graph obtained in
Step 3. This results in a vertical shift up if
or down if
.
16
MA100,Week 3
Let
and be functions with domains
, and
are defined as follows:
Transformations, Operations on Functions, Inverses
and . Then the functions
17
MA100,Week 3
Example Given
Solution
Transformations, Operations on Functions, Inverses
and
Domain:
2
18
MA100,Week 3
Transformations, Operations on Functions, Inverses
Definition Given two functions
called the composition of and
x
and , the composite function
) is defined by
g
g(x)
f(g(x))
f
input
The domain of
is defined whenever both
(also
output
and
are defined
19
MA100,Week 3
Example
Suppose that
Find
Solution
.
Transformations, Operations on Functions, Inverses
2
2
20
MA100,Week 3
Example Suppose that
Find
Solution
2
2
Transformations, Operations on Functions, Inverses
2
2
21
MA100,Week 3
Transformations, Operations on Functions, Inverses
2
Example Let
and
Find
and its domain
Solution The domain of is
.
Since is everywhere defined, the domain of
is also
.
2
2
2
22
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Finding Formulas for Composites
If
and
, find
(a)
(b)
(c)
(d)
Solution
(a)
Domain
23
MA100,Week 3
Transformations, Operations on Functions, Inverses
(a)
is defined for all real
Domain
24
MA100,Week 3
(b)
and
Transformations, Operations on Functions, Inverses
Domain
25
MA100,Week 3
Transformations, Operations on Functions, Inverses
and
(c)
1/4
Domain
26
MA100,Week 3
(d)
Transformations, Operations on Functions, Inverses
and
Domain
27
MA100,Week 3
Transformations, Operations on Functions, Inverses
The composition of functions can be extended to include the composition of
three or more functions.
The composite function
is found by applying
and,
28
MA100,Week 3
Transformations, Operations on Functions, Inverses
Theorem Properties of Function Composition:
Suppose
, and
are functions.

 If is defined as
, provided the composite functions are defined.
for all real numbers , then
.
29
MA100,Week 3
Example Find
Solution
if
Transformations, Operations on Functions, Inverses
10
, and
.
10
10
10
30
MA100,Week 3
Example Find
Solution
Transformations, Operations on Functions, Inverses
10
if
, and
.
10
10
10
31
MA100,Week 3
Transformations, Operations on Functions, Inverses
A function is called a one-to-one function if it never takes on the same value
twice; that is,
1
2
1
2
Functions that share this property with are called one-to-one functions.
32
MA100,Week 3
Transformations, Operations on Functions, Inverses
HORIZONTAL LINE TEST
A function is one-to-one if and only if no horizontal line intersects its graph
more than once.
3
No horizontal line intersects the graph
more than once.
Therefore
3
If
1
2 , then
3
1
3
2
By Definition,
3
is one-to-one.
33
MA100,Week 3
Transformations, Operations on Functions, Inverses
Horizontal Line Test
2
There are horizontal lines that
intersect the graph more than
once
2
1 and -1 have the same output
2 is not one-to-one.
34
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Determine whether the function is one-to-one.
Solution
1
2
1
2
This function is one-to-one
1
2
1
2
35
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Determine whether the function is one-to-one.
2
Solution
The graph of
axis of symmetry
Pick
𝑏
2𝑎
values equidistant from 2
2
is a parabola
4
2(−1)
The function is not one-to-one
36
MA100,Week 3
Transformations, Operations on Functions, Inverses
Definition
Let be a one-to-one function with domain and range . Then its inverse
function −1 has domain and range and is defined by
for any
in .
−1
−1
−1
37
MA100,Week 3
maps
−1
into
A
Transformations, Operations on Functions, Inverses
maps
back into
f
B
x
𝑦 = 𝑓(𝑥)
−1
Note:
−1
does not mean
1
𝑓(𝑥)
−1
reverses the effect of .
−1
1
𝑓(𝑥)
38
MA100,Week 3
Example If
Solution
Transformations, Operations on Functions, Inverses
is a one-to-one function such that
−1
−1
, what is
−1
?
−1
−1
−1
39
MA100,Week 3
Transformations, Operations on Functions, Inverses
We usually reverse the roles of
and
and write
−1
Let
be a one-to-one function with domain and range . The inverse function
−1
satisfies the following cancellation properties.
−1
Conversely, any function
−1
−1
satisfying these equations is the inverse of .
−1
undoes what
undoes what
does.
−1 does.
40
MA100,Week 3
3 and
Example Show that
Solution
Note that the domain and range of both
Thus,
and
Transformations, Operations on Functions, Inverses
1/3 are inverses of each other.
and
is . We have
3
3 1/3
1/3
are inverses of each other.
1/3 3
Cube function and the cube root function, when composed, cancel each other.
41
MA100,Week 3
Transformations, Operations on Functions, Inverses
How to Find the Inverse of a One-to-One Function
1 Write
.
2 Solve this equation for in terms of (if possible).
−1
3 Interchange and . The resulting equation is
.
42
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Find the inverse of the function
Solution We first write
5
2
.
and solve for .
5
5
Then we interchange
5
𝑥5 −3
and
to get
−1
1/5
1/5
. The inverse function is
1/5
43
MA100,Week 3
The graph of
Suppose that
−1
−1
−1
Transformations, Operations on Functions, Inverses
is obtained by reflecting the graph of
about the line
.
is any point on the graph of a function
is on the graph of −1
cancellation properties
44
MA100,Week 3
Transformations, Operations on Functions, Inverses
Example Sketch the graph of
Use the graph of to sketch the graph of
Find an equation for −1 .
Solution
Sketch the graph
Reflect the graph of
about the line
.
−1
.
to get
−1
45
MA100,Week 3
Solve
Transformations, Operations on Functions, Inverses
for , noting that
Square each side
Add 2
2
Interchange
.
2
and
−1
2
2
46
MA100,Week 3
Transformations, Operations on Functions, Inverses
Theorem
Properties of Inverse Functions:
Suppose
and
 The range of
are inverse functions.
is the domain of

if and only if

is on the graph of
and the domain of
if and only if
is the range of
is on the graph of
47