Chapter 1
Functions
1
Overview
• Definition
• Domain and Range
• Rates of Change
• Composition
• Transformations
• Absolute Value
• Inverses
2
Chapter 1.1
What is a Function?
3
A Function
• A rule, or relation, that goes from x to y, or x to f(x)
• Must be unique – for each value of x, have only one f(x)
• Eliminates confusion – for each input there is a unique output
4
Function or Not?
5
Clearly a function
• Each value of the argument, p, has a unique partner h(p)
(Note, the variables needn’t be x and y!)
6
Clearly a function
• Can also use vertical line test!
Any vertical line can only intersect the function curve once.
7
Function or Not?
8
• A and B are functions
• C is not. For x = 0, y can be ±3
• Again, vertical line test
9
Another non-function relationship
• Input: the first name of each of you
• Output: the last name of each of you
10
Why do we care?
• Many mathematical principles are confused if we apply them
to non-functions
• Unique results, for example, can be very important
11
Evaluating a Function
• Consider the function f(x) = (x2 - 45)/3
• Evaluate f(x) for x = 9
12
Solution
• Consider the function f(x) = (x2 - 45)/3
• Evaluate f(x) for x = 9
• Put 9 everywhere there is an x:
f(9) = (92 – 45)/3 = (81-45)/3 = 36/3 = 4
• f(9) = 4
13
One to One Functions
• Each output value corresponds to only one input value
• Examples of functions that are not one-to-one:
f(x) = 5
f(x) = x2
f(x) = sin(x)
14
A Test
• A one-to-one function meets the horizontal line test
• Here, f(0)= f(2) = 1; not one-to-one
15
Example
• Is this a function? Is it one-to-one?
16
Solution
• It is not a function
17
Example
• Is this a function? Is it one-to-one?
18
Solution
• It is a function
• It is not one-to-one
19
• Is this a function? Is it one-to-one
20
Solution
• This is a one-to-one function
21
Summary
• A function assigns a unique value, f(x) to every input value, x
• A one-to-one function is a function that has a unique input x,
for every output f(x)
22
Chapter 1.2
Domain and Range
23
Overview
• Domain and range
• Piece-wise functions
24
Domain and Range
• Domain is the values that go into the function: you live in your
domain
• Range is the values that come out of the function: you run
around in your range
25
Domain and Range
• Specifying a domain may
– Exclude points for which a function is not defined
– Restrict the input values such that we have a function
26
Example
• What are the domain and range of:
a. { (2, 5), (4, 8,), (1, 2)}
b. {(4, 5), (4, 6), (1, -5)}
Are they functions?
27
Solution
• What are the domain and range of
a. { (2, 5), (4, 8), (1, 2)}
b. {(4, 5), (4, 6), (1, -5)}
a: Domain {2, 4, 1}, Range{5, 8, 2} yes a function
b: Domain {4, 1} Range {5, 6, -6} no, not a function;
for an input of 4 have two different outputs
28
Notation
• Unfortunately, since there are so many applications of
mathematics, there are many notations used to represent
domain and range
• Use whatever notation you are comfortable with, however,
you should learn to understand the rest
• Whatever you do – don’t panic over notation! Use words, if
required. Many famous mathematicians, for centuries, have
done so
29
Some Notation
• We can express range and domain in multiple ways
– Number line: open circle means point is not included
– Intervals: “(“ means point is not included
[1,3] ∪ 5,7
– Inequalities: 1 ≤ 𝑥 ≤ 3, 𝑎𝑛𝑑 5 < 𝑥
• How would you express the following line in interval notation?
-10
-0
10
30
Solution
• Notation: (-10, ∞)
-10
-0
10
31
Yet one more…
• Set notation:
{x | 10 < x }
• Reads as: the set of values for x such that 10 is less than x
32
Functions to Watch For
• Rational Functions:
4
5−𝑥
• Square Roots: 1 − 5𝑥
• Logarithms: log (1-x)
Each of these functions has restricted domains
33
Constant Function
34
Identity Function
• f(x) = x
35
Absolute Value
f(x) = |x|
36
Quadratic: f(x) = x2
37
Cubic: f(x) = x3
38
Reciprocal: f(x) = 1/x
39
Square Root: f(x) = 𝑥
40
Piecewise Functions
• Quite simply, functions that are defined in pieces
41
Example: Graph
Note open and closed circles!
42
Example
• Determine if the following is a function, then graph and give
the range and domain of:
43
Solution
This is a function.
Domain is x≠ 0
Range is all y
44
Find the range and domain
• f (x) = 7 − 𝑥
• 𝑓 𝑥 =
2
1−𝑥
• 𝑓 𝑥 =𝑐
• 𝑓 𝑥 =𝑥
• 𝑓 𝑥 = 𝑥3
45
Solutions
• f (x) = 7 − 𝑥 :x ≤ 7, 𝑦 ≥ 0
• 𝑓 𝑥 =
2
1−𝑥
: 𝑥 ≠ 1, 𝑦 ≠ 0
• 𝑓 𝑥 = 𝑐 : x = all numbers, y = c
• 𝑓 𝑥 = 𝑥 : x = all numbers, y = all numbers
• 𝑓 𝑥 = 𝑥 3 : x = all numbers, y = all numbers
46
Examples
Find the domain of
y = 3𝑥 + 9
1
y= 2
𝑡 −6𝑡 −7
47
Solution
Find the domain of
y = 3𝑥 + 9
If we are looking for y to be a real number, we need to
have 3x + 9 ≥ 0. Our domain is x ≥ -3
1
y= 2
𝑡 −6𝑡 −7
We need to avoid dividing by zero, so our domain is
all real numbers, excluding 7 and -1
48
Example
Find the domain of
𝑥+3
𝑥 −4
3
𝑥+3
𝑥 −4
49
Solution
• For both of these expressions, we need x ≠4
• For the first, we also want the value under the square root
to be ≥ 0. This restricts x to be x ≤ -3 υ x > 4
𝑥+3
𝑥 −4
• For the second, there is no restriction that the value be ≥ 0, so
we have only x ≠4
3
𝑥+3
𝑥 −4
50
Finding the Range
Find the range of
y=
𝑥+2
𝑥 −3
51
Solution
Solve for x:
𝑥+2
y=
𝑥 −3
y(x-3) = x + 2
xy – x = 3y + 2
3𝑦+2
x=
𝑦 −1
We can see that for any y there is a x, excluding y = 1. However,
if y = 1 is a solution, that means 2 = -3, which is impossible
The range is all reals excluding y = 1
NOTE: This technique does not always work!
52
If can graph, can find domain and range
53
Some Functions You Will Need to Know
y=|x|
y=x2
y= 𝑥
y=x3
y=1/x
y= 1 − 𝑥 2
54
Practice
• Try the exercises in section 1.2
• Bring “problem children” to class tomorrow
55
Chapter 1.3
Rates of Change and Behavior of Graphs
56
Overview
• Average Rate of Change
• Increasing and decreasing functions
• Local and absolute maxima and minima
57
Rate of Change
• Change in output / change in input
• Change in domain/ change in range
• Change in y / change in x
• Change in f(x) / change in x
•
∆𝑦
∆𝑥
• (y1-y2) / (x1-x2)
• ( f(x1)- f(x2)) / (x1-x1)
• What is another term for rate of change?
58
• Rate of change is slope!
59
Example
• Find the average rate of change of
f(x) = 3x + 4
on the interval (-3, 4)
60
Solution
• Find the average rate of change of
f(x) = 3x2 + 4
on the interval (-3, 4)
• f(-3) = 9 + 4 = 12
• f(4) = 48 + 4 = 52
• f(3) – f(4) = 12-52 = 40
• -3 – (4) = -7
• Average rate of change is -40 / 7
61
Example
• Find the average rate of change of
f(x) = x3 – x2 + 4
on the interval (0, a)
62
Solution
• Find the average rate of change of
f(x) = x3 – x2 + 4
on the interval (0, a)
• f(0) =4
• f(a) = a3 – a2 + 4
• Rate of change is
𝑎 3 − 𝑎2
𝑎
=
a2 – a
63
Find the Average Rate of Change
• p(x) = 3x + 4 on [2, 2+h]
64
Solution
• p(x) = 3x + 4 on [2, 2+h]
p(2) = 6+4=10, p(2+h) = 6 + 4 + 3h = 6+ 4 + 3h
p(2) – p(2+h) = -3h
2 – (2+h) = -h
Average rate of change = 3
65
Increasing and Decreasing Functions
•
The function 𝑓(𝑥) = 𝑥3 − 12𝑥 is
increasing on (−∞, −2) ∪ (2, ∞) and
is decreasing on (−2, 2).
• Increasing means rate of
change > 0
• Decreasing means rate of
change< 0
66
Maxima and Minima
Local Max
Local maximum at b means f(b) ≥f(x)
for x in the interval (a, c) and a < b< c
Local minimum at b means f(b) ≤ f(x)
for x in the interval (a, c) and a < b< c
Local maximum at x = = - 2, y = 16
Local minimum at x = 2, y = -16
Local Min
67
Formal Definitions
• A function f is an increasing function on an open interval if f (b) > f (a)
for any two input values a and b in the given interval where b > a
• A function f is a decreasing function on an open interval if f (b) < f (a)
for any two input values a and b in the given interval where b > a
• A function f has a local maximum at x = b if there exists an interval
(a, c) with a < b < c such that, for any x in the interval (a, c), f (x) ≤ f (b)
• Likewise, f has a local minimum at x = b if there exists an interval
(a, c) with a < b < c such that, for any x in the interval (a, c), f (x) ≥ f (b)
68
Example
• Where is this function increasing or decreasing?
• Are there local minima, maxima?
69
Solution
• Increasing 1<x<3 and x>4
• Decreasing x<1 and 3<x<4
• Local Minima:
(1,-1), (4, ~0.6)
• Local Maxima
(3, 1)
70
Absolute Maximum and Minimum
The absolute maximum of f at x = c is f (c) where f (c) ≥ f (x)
for all x in the domain of f .
The absolute minimum of f at x = d is f (d) where f (d) ≤ f (x)
for all x in the domain of f
71
Chapter 1.4
Composition of Functions
72
Overview
• Combine functions using algebra
• Composition of functions
• Evaluating compositions
• Domain of composed functions
• Decomposing a composed function
73
Arithmetic Operations
• (f + g)(x) = f(x) + g(x)
• (f - g)(x) = f(x) - g(x)
• (fg)(x) = f(x)g(x)
• (f / g)(x) = f(x) / g(x), for g(x) ≠ 0
• The domain is all inputs belonging to both f and g, with the
added provision that in the division g(x) ≠ 0
74
Examples
If f(x) = x and g(x) = x - 1
• (f + g)(x) = f(x) + g(x) = 2x -1
• (f – g)(x) = f(x) – g(x) = 1
• (fg)(x) = f(x)g(x) = x2 – x
• (f/g)(x) =f(x)/g(x) =
𝑥
𝑥−1
for x ≠ 1
We can also include constants:
• (f + 2g)(x) = f(x) + 2g(x) = 3x - 2, for example
75
Composition of Functions
If f(x) = 3x and g(x) = x - 1
• f(g(x)) = 3(x-1)
We also write this as f  g(x); use g(x) as the input to f
The domain of f  g(x) is those inputs in the domain of g for
which g(x) is in the domain of f
Note: In general, f  g(x) ≠ g  f(x)
We say f composed with g or, simply, f of g
76
Examples
f(x) = x2 and g(x) = 2x + 2
Find f  g(x) and g  f(x)
77
Solution
f(x) = x2 and g(x) = 2x + 2
Find f  g(x) and g  f(x)
f  g(x) = (2x+2) 2: Domain is all numbers, range is y4
g  f(x) = 2x2 + 2 : Domain is all numbers, range is y 2
78
Example
f(x) = x2 + 3 and g(x) = 𝑥
What is the domain of f(g(x))?
79
Solution
• f(x) = x2 + 3 and g(x) = 𝑥
• f(g(x)) = x +3, but the domain is not all real numbers.
• The domain of g is all positive reals,
hence the domain of f(g(x))is all numbers >3;
it is not the same as the domain of f(x)
80
More on Domain and Composition
𝑓 𝑥 =
1
𝑥+2
and 𝑔 𝑥 =
𝑥
𝑥−3
81
Continuing: Domain of f(g(x))
𝑓 𝑥 =
1
𝑥+2
and 𝑔 𝑥 =
𝑥
𝑥−3
3 is not in the domain of g(x), so it is not in the domain of f(g(x))
Additionally, we can’t have f(-2), so g(x)  -2.
For g(x) = -2, we need -2 =
𝑥
𝑥−3
or x = 2
So, the domain of f(g(x)) is all reals, but x  3 and x  2
82
What about g(f(x))?
𝑓 𝑥 =
1
𝑥+2
and 𝑔 𝑥 =
𝑥
𝑥−3
• To satisfy f, x  2
• To satisfy g, f(x)  3
for f(x) = 3, we have 3 =
1
𝑥+2
, 3x + 6 = 1
Thus, x  -5/3
• Domain is x  2, x  -5/3
83
More Examples
Find f(g(x)) and g(f(x)) and the domain of each
• f(x) = 𝑥 − 2, g(x) =
• f(x) =
𝑥−3
2𝑥
, g(x) =
1
3−𝑥
3
𝑥
84
Solution
Find f(g(x)) and g(f(x)) and the domain of each
• f(x) = 𝑥 − 2, g(x) =
f(g(x)) =
1
3−𝑥
1
3−𝑥
− 2 : x≠ 3 and 1/(3-x) ≥ 2; the second gives
3-x ≤ ½, or x ≥ 5/2
g(f(x))=
1
3− 𝑥−2
; need x≥ 2 and x≠ 11
85
Solution
Find f(g(x)) and g(f(x)) and the domain of each
• f(x) =
𝑥−3
2𝑥
, g(x) =
3
𝑥
f(g(x)) = [3/x – 3] / [6/x] = (3 – 3x)/6 = (1 – x)/2
can’t have x = 0 because of the definition of g.
Range is x≠ 0
g(f(x)) = 3/[(x-3)/2x] = 6x/(x-3)
can’t have x = 0 because of the definition of x, also
cannot have x=3
Range is (−∞, 0) ∪ (0,3) ∪ (3, ∞)
86
Example
• A store offers customers a 30% discount on the price x of
selected items. Then, the store takes off an additional 15% at
the cash register.
Write a price function P(x) that computes the final price of the
item in terms of the original price x.
87
Solution
• A store offers customers a 30% discount on the price x of
selected items. Then, the store takes off an additional 15% at
the cash register.
Write a price function P(x) that computes the final price of the
item in terms of the original price x.
Final price = register discount (price discount (item))
FP(x) = 0.85 ( 0.7 (x))
(0.85)(0.7) = 0.595
FP(x) = 0.595 x
88
Summary
• When composing functions, f(g(x)), the domain is only
those values where g(x) is defined and f(y), where y = g(x),
are defined
BE CAREFUL
• f(a+b)  f(a) + f(b)
• f(ab)  f(a)(f(b)
• f(a/b)  f(a)/f(b)
• f(g(x))  g(f(x))
89
Chapter 1.5
Transformation of Functions
90
Overview
• Vertical and horizontal shift
• Even and odd functions
• Compression and stretching
• Transformation combinations
91
Vertical Shift
•
G(x) = f(x)+k
•
Here we have the vertical shift by 𝑘 = 1 of the cube root function 𝑓(𝑥) =
3
𝑥.
92
Horizontal Shift
•
G(x) = f(x+k) shifts the function k units to the left
•
Horizontal shift of the function 𝑓(𝑥) =
•
Note that ℎ = + 1 shifts the graph to the left, that is, towards negative values of x.
3
𝑥.
93
How to do a shift?
1. Use a table, as when you graph a function
2. Move the function on the graph itself
For example, f(x) = 4x
x
f(x)
-2
-1
0
1
2
f(x+3)
-8
-4
0
4
8
4
8
12
16
20
25
20
f(x)+3
-5
-1
3
7
11
15
f(x)
10
f(x+3)
5
f(x)+3
0
-4
-2
-5
0
2
4
-10
94
Example
• For f(x) = x2 + 3, graph f(x), f(x+2), f(x)+2
95
Solution
• For f(x) = x2 + 3, graph f(x), f(x+2), f(x)+2
x
f(x)
-2
-1
0
1
2
f(x+2)
7
4
3
4
7
0
1
4
9
16
f(x)+2
9
6
5
6
9
96
Combining Horizontal and Vertical Shifts
• Suppose f(x) = |x|, h(x) = f(x+1) + 3. Graph h(x)
97
Solution
x
f(x)
-3
-2
-1
0
1
2
3
f(x+1)
3
2
1
0
1
2
3
h(x)
2
1
0
1
2
3
4
5
4
3
4
5
6
7
98
Reflections
• Given a function f (x), a new function g(x) = − f (x) is a vertical reflection of
the function f (x), sometimes called a reflection about the x-axis.
• Given a function f (x), a new function g(x) = f ( − x) is a horizontal reflection
of the function f (x), sometimes called a reflection about the y-axis.
99
Example
• f(x) = x2; graph g(x) = -f(x), h(x) = f(-x)
100
Solution
• f(x) = x2; graph g(x) = -f(x), h(x) = f(-x)
Note f(x) = h(x)!
x
f(x)
-3
-2
-1
0
1
2
3
g(x)
9
4
1
0
1
4
9
h(x)
-9
-4
-1
0
-1
-4
-9
9
4
1
0
1
4
9
101
Example
Sketch the following, starting with y = x2 :
y = x2 + 3
y = (x-4) 2 + 3
y = -(x-4) 2 - 3
102
Solution
Sketch the following, starting with y = x2 :
y = x2 + 3
y = (x-4) 2 + 3
y = -(x-4) 2 - 3
103
Example
• Use the graph y = |x| to graph y = - |x - 5| - 4
104
Solution
• Use the graph y = |x| to graph y = - |x - 5| - 4
105
Be careful with reflection and sign!
• Graph y = −𝑥 + 2 ; So far we have looked at y = f(x+c)
• Start with y = 𝑥 and shift to the left 2 units, y = 𝑥 + 2
•
Replace x by –x and reflect the graph in the x axis
106
Vertical Scaling (Compression and Stretching)
• If f is a function, and c a constant >0, replacing y by cy gives
y = 1/c f(x)
If c > 1, the graph is compressed by a factor of c
If c < 1, the graph is stretched by a factor of c
107
Vertical Scaling Example
• Start with y = 2x – x2, 0 ≤ x ≤ 2
• Now, consider 2y = 2x – x2 or y = ½(2x - x2)
• Now, consider 1/2y = 2x – x2
108
Horizontal Scaling
• y = f(x) goes to y = f(cx)
• Compare y = x2 with y = (3x)2 and y = (x/3) 2
109
Solution
Compare y = x2 with y = (3x)2 and y = (x/3) 2
x
x*x
-3
-2
-1
0
1
2
3
9
4
1
0
1
4
9
(3x)*(3x)
81
36
9
0
9
36
81
(x/3)*(x/3)
1
0.4444444
0.1111111
0
0.1111111
0.4444444
1
40
30
20
10
0
-4
-2
x*x
0
(3x)*(3x)
2
4
(x/3)*(x/3)
110
Transformation Summary
• Let (a,b) be on the graph of y = f(x) and c a positive constant
Transformation
Change
New Equation
New Point
Shift up c units
y y – c
y – c = f(x)
(a, b+c)
Shift down c units
y y + c
y + c = f(x)
(a, b-c)
Shift right c units
x x – c
y = f(x-c)
(a+c, b)
Shift left c units
y x + c
y = f(x+c)
(a-c, b)
Reflection in x-axis
y  -y
y = -f(x)
(a, -b)
Reflection in y-axis
x  -x
y = f(-x)
(-a, b)
Vertical Scale Change
Horizontal Scale Change
y  cy
x  cx
y = f(x)/c
y =f(cx)
(a, b/c)
(a/c, b)
111
Using Transformation
Graph y = 4(x-5)2 -3
• Start with y = x2
• Vertically stretch y = 4x2
• Replace x by x-5
• Replace y by y+3
There is no unique procedure/order!
112
Solution
Graph y = 4(x-5)2 -3
113
Graph using Transformation
• y = 1/(5-x)
• y= 𝑥−3+1
• y = - 1 − (𝑥 − 2)2
114
y = 1/(5-x)
Blue is basic 1x
Red is 1/(-x)
115
y= 𝑥−3+1
Blue: Initial
Red: H shift 3
Green: V shift 1
116
y = - 1 − (𝑥 − 2)2
Blue:
1 − 𝑥2
Red: - 1 − 𝑥 2
Green: Final
117
How do we find a transformation from equation?
• Suppose we had
y = 1/3 f(-x/2 + 4) + 5
118
Solution
Finding a transformation from an equation
• Suppose we had
y = 1/3 f(-x/2 + 4) + 5
• If we start with y = f(x)
– Horizontal scale change of ½ -- graph is stretched by a
factor of two
– Replace x by -x, reflection on y axis
– Substitute x-8 for x, creating a shift to the right
– Substitute 3y for y, a vertical scale change (compression)
– Now shift the graph up by 5 units
119
And a Point?
If (a,b) was on the graph of y = f(x), what is that point on
y = 1/3 f(-x/2 + 4) + 5
• (a,b)  (2a, b)  (-2a, b) (-2a – 8,b) (-2a – 8, b/3)
(-2a+8, b/3 + 5)
120
Even and Odd Functions
• Even functions: f(-x) = f(x)
• Odd functions: f(-x) = -f(x)
• For a function to be even, the rule must hold for each value in
the domain
• Same with an odd function
• There can be no points of exceptions; a function is even, odd,
or neither
• The only function that is both even and odd is f(x)=0
121
Even, Odd or Neither?
• f(x) =
• g(x) =
1 − 𝑥2
2+ 𝑥 2
𝑥 − 𝑥3
2𝑥+ 𝑥 3
• h(x) = (x2 + x) 2
• f(x) = x |x| /
𝑥 −4
1 if x > 0
• g(x) =
0 if x = 0
-1 if x < 0
122
Solution
• f(x) =
• g(x) =
1 − 𝑥2
2+ 𝑥 2
even
𝑥 − 𝑥3
2𝑥+ 𝑥 3
even
• h(x) = (x2 + x) 2
neither
• f(x) = x |x| /
odd
𝑥 −4
1 if x > 0
• g(x) =
0 if x = 0
odd
-1 if x < 0
123
Recap
• We can create graphs of many functions by graphing
canonical functions and then modifying them
• Some functions you should know:
 y = x2
 y= 𝑥
 y = 1 − 𝑥2
 y = 𝑥3
 y = |x|
 y = 1/x
124
Modification Techniques
• Horizontal shift: x becomes x ± a
• Vertical shift: y becomes y ± a
• Compression/expansion: x or y becomes cx or cy
• Reflection in x or y axis
125
General Rules
• When combining vertical transformations written in the form a
f (x) + k, first stretch by a and then shift by k
• When combining horizontal transformations written in the form
f (bx + h), first shift by h and then stretch by b
• When combining horizontal transformations written in the form
f (b(x + h)), first horizontally stretch by b and then horizontally
shift by h
• Horizontal and vertical transformations are independent. It
does not matter whether horizontal or vertical transformations
are performed first.
126
Key things to watch for:
1. Watch out when going from –x to –x – a or –x + a
Example: creating y =
1
−𝑥+𝑎
– Either reflect x to get –x before shifting by a or reflect x – a
to get –x + a
2. Be careful of f(ax – c)
May be best to work with f(a[x -
𝑐
]
𝑎
)
127
General Rule, Again
• When combining horizontal transformations written in the form
• f (bx + h), first shift by h and then stretch by b
• When combining horizontal transformations written in the form
f (b(x + h)), first horizontally stretch by b and then horizontally
shift by h
• Think about the parentheses! Does the transform include the
whole argument? If so, do that first!
• When in doubt, make a table of key points!
128
Chapter 1.6
Absolute Value
129