Functions and Their Graphs
Jackie Nicholas
Janet Hunter
Jacqui Hargreaves
Mathematics Learning Centre
University of Sydney
NSW 2006
1999
c University of Sydney
,Mathematics Learning Centre, University of Sydney i
Contents
1 Functions 1
1.1 What is a function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Definition of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Vertical Line Test . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Domain of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Range of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Specifying or restricting the domain of a function . . . . . . . . . . . . . . 6
1.3 The absolute value function . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 More about functions 11
2.1 Modifying functions by shifting . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Vertical shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Horizontal shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Modifying functions by stretching . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Modifying functions by reflections . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Reflection in the x-axis . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Reflection in the y-axis . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Other effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Combining effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Graphing by addition of ordinates . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Using graphs to solve equations . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Even and odd functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.10 Increasing and decreasing functions . . . . . . . . . . . . . . . . . . . . . . 23
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Piecewise functions and solving inequalities 27
3.1 Piecewise functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Restricting the domain . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
,Mathematics Learning Centre, University of Sydney ii
4 Polynomials 36
4.1 Graphs of polynomials and their zeros . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Behaviour of polynomials when |x| is large . . . . . . . . . . . . . . 36
4.1.2 Polynomial equations and their roots . . . . . . . . . . . . . . . . . 37
4.1.3 Zeros of the quadratic polynomial . . . . . . . . . . . . . . . . . . . 37
4.1.4 Zeros of cubic polynomials . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Polynomials of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Factorising polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.1 Dividing polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.2 The Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4.3 The Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Solutions to exercises 50
, Mathematics Learning Centre, University of Sydney 1
1 Functions
In this Chapter we will cover various aspects of functions. We will look at the definition of
a function, the domain and range of a function, what we mean by specifying the domain
of a function and absolute value function.
1.1 What is a function?
1.1.1 Definition of a function
A function f from a set of elements X to a set of elements Y is a rule that
assigns to each element x in X exactly one element y in Y .
One way to demonstrate the meaning of this definition is by using arrow diagrams.
X Y X Y
f g
1 5 1 5
2 2 6
3 3 3 3
4 2 4 2
f : X → Y is a function. Every element g : X → Y is not a function. The ele-
in X has associated with it exactly one ment 1 in set X is assigned two elements,
element of Y . 5 and 6 in set Y .
A function can also be described as a set of ordered pairs (x, y) such that for any x-value in
the set, there is only one y-value. This means that there cannot be any repeated x-values
with different y-values.
The examples above can be described by the following sets of ordered pairs.
F = {(1,5),(3,3),(2,3),(4,2)} is a func- G = {(1,5),(4,2),(2,3),(3,3),(1,6)} is not
tion. a function.
The definition we have given is a general one. While in the examples we have used numbers
as elements of X and Y , there is no reason why this must be so. However, in these notes
we will only consider functions where X and Y are subsets of the real numbers.
In this setting, we often describe a function using the rule, y = f (x), and create a graph
of that function by plotting the ordered pairs (x, f (x)) on the Cartesian Plane. This
graphical representation allows us to use a test to decide whether or not we have the
graph of a function: The Vertical Line Test.
Jackie Nicholas
Janet Hunter
Jacqui Hargreaves
Mathematics Learning Centre
University of Sydney
NSW 2006
1999
c University of Sydney
,Mathematics Learning Centre, University of Sydney i
Contents
1 Functions 1
1.1 What is a function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Definition of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Vertical Line Test . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Domain of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Range of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Specifying or restricting the domain of a function . . . . . . . . . . . . . . 6
1.3 The absolute value function . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 More about functions 11
2.1 Modifying functions by shifting . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Vertical shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Horizontal shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Modifying functions by stretching . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Modifying functions by reflections . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Reflection in the x-axis . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Reflection in the y-axis . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Other effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Combining effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Graphing by addition of ordinates . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Using graphs to solve equations . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Even and odd functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.10 Increasing and decreasing functions . . . . . . . . . . . . . . . . . . . . . . 23
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Piecewise functions and solving inequalities 27
3.1 Piecewise functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Restricting the domain . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
,Mathematics Learning Centre, University of Sydney ii
4 Polynomials 36
4.1 Graphs of polynomials and their zeros . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Behaviour of polynomials when |x| is large . . . . . . . . . . . . . . 36
4.1.2 Polynomial equations and their roots . . . . . . . . . . . . . . . . . 37
4.1.3 Zeros of the quadratic polynomial . . . . . . . . . . . . . . . . . . . 37
4.1.4 Zeros of cubic polynomials . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Polynomials of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Factorising polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.1 Dividing polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.2 The Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4.3 The Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Solutions to exercises 50
, Mathematics Learning Centre, University of Sydney 1
1 Functions
In this Chapter we will cover various aspects of functions. We will look at the definition of
a function, the domain and range of a function, what we mean by specifying the domain
of a function and absolute value function.
1.1 What is a function?
1.1.1 Definition of a function
A function f from a set of elements X to a set of elements Y is a rule that
assigns to each element x in X exactly one element y in Y .
One way to demonstrate the meaning of this definition is by using arrow diagrams.
X Y X Y
f g
1 5 1 5
2 2 6
3 3 3 3
4 2 4 2
f : X → Y is a function. Every element g : X → Y is not a function. The ele-
in X has associated with it exactly one ment 1 in set X is assigned two elements,
element of Y . 5 and 6 in set Y .
A function can also be described as a set of ordered pairs (x, y) such that for any x-value in
the set, there is only one y-value. This means that there cannot be any repeated x-values
with different y-values.
The examples above can be described by the following sets of ordered pairs.
F = {(1,5),(3,3),(2,3),(4,2)} is a func- G = {(1,5),(4,2),(2,3),(3,3),(1,6)} is not
tion. a function.
The definition we have given is a general one. While in the examples we have used numbers
as elements of X and Y , there is no reason why this must be so. However, in these notes
we will only consider functions where X and Y are subsets of the real numbers.
In this setting, we often describe a function using the rule, y = f (x), and create a graph
of that function by plotting the ordered pairs (x, f (x)) on the Cartesian Plane. This
graphical representation allows us to use a test to decide whether or not we have the
graph of a function: The Vertical Line Test.
