Chapter 6 focuses on the concepts of functions, relations, and transformations in mathematics. It covers essential topics such as set notation, the definition of relations, and the characteristics of functions, including one-to-one functions. The chapter also explores transformations like translations, dilations, and reflections, providing a comprehensive understanding of how these concepts apply to graphing and problem-solving. This content is particularly useful for students studying algebra and calculus, offering practical examples and exercises to reinforce learning.

Key Points

  • Explains set notation and its application in mathematical contexts.
  • Defines relations and functions, emphasizing the concept of one-to-one functions.
  • Covers transformations including translations, dilations, and reflections.
  • Includes practical examples and exercises for reinforcing mathematical concepts.
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9780521740524c06.xml CUAU021-EVANS August 22, 2008 22:4
CHAPTER
6
Functions, Relations
and Transformations
Objectives
To understand and use the notation of sets, including the symbols ∈, ⊆, ∩, ∪,Ø
and \.
To use the notation for sets of numbers.
To understand the concept of relation.
To understand the terms domain and range.
To understand the concept of function.
To understand the term one-to-one.
To understand the terms implied (maximal) domain, restriction of a function and
hybrid function.
To be able to find the inverse of a one-to-one function.
To define dilations from the axes, reflections in the axes and translations.
To be able to apply transformations to graphs of relations.
To apply a knowledge of functions to solving problems.
Sections 6.1 and 6.2 of this chapter introduce the notation that will be used throughout the rest
of the book. You will have met much of it before and this will serve as revision. The language
introduced in this chapter helps to express important mathematical ideas precisely. Initially
they may seem unnecessarily abstract, but later in the book you will find them used more and
more in practical situations.
6.1 Set notation and sets of numbers
Set notation
Set notation is used widely in mathematics and in this book where appropriate. This section
summarises all of the set notation you will need.
163
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164 Essential Mathematical Methods1&2CAS
By a set we mean a collection of objects. The objects that are in the set are known as
elements or members of the set. If x is an element of a set A we write x ∈ A. This can also be
read as ‘x is a member of the set of A’or‘x belongs to A’or‘x is in A’.
The notation x /∈ A means x is not an element of A.For example:
2 /∈ set of odd numbers
Set B is called a subset of set A, if and only if
x ∈ B implies x ∈ A
To indicate that B is a subset of A,wewrite B ⊆ A.
This expression can also be read as ‘B is contained in A’or‘A contains B’.
The set of elements common to two sets A and B is called the intersection of A and B and is
denoted by A ∩ B. Thus x ∈ A ∩ B if and only if x ∈ A and x ∈ B.
If the sets A and B have no elements in common, we say A and B are disjoint, and write
A ∩ B = Ø. The set Ø is called the empty set.
The union of sets A and B, written A ∪ B,isthe set of elements that are either in A or in B.
This does not exclude objects that are elements of both A and B.
Example 1
For A = {1, 2, 3, 7} and B = {3, 4, 5, 6, 7}, find:
a A ∩ B b A ∪ B
Solution
a A ∩ B = {3, 7} b A ∪ B = {1, 2, 3, 4, 5, 6, 7}
In Example 1, 3 ∈ A and 5 /∈ A and {2, 3} ⊆ A.
Finally we introduce the set difference of two sets A and B:
A \ B ={x: x ∈ A, x /∈ B}
A \ B is the set of elements of A that are not elements of B.For sets A and B in Example 1,
A \ B = {1, 2} and B \ A = {4, 5, 6}.
Sets of numbers
We begin by recalling that the elements of {1, 2, 3, 4, . . .} are called the natural numbers and
the elements of {...,−2, −1, 0, 1, 2,...} are called integers.
The numbers of the form
p
q
, with p and q integers, q = 0, are called rational numbers.
The real numbers which are not rationals are called irrational (e.g. and
√
2).
The rationals may be characterised by the property that each rational number may be written
as a terminating or recurring decimal.
The set of real numbers will be denoted by R.
The set of rational numbers will be denoted by Q.
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Chapter6—Functions, Relations and Transformations 165
The set of integers will be denoted by Z.
The set of natural numbers will be
denoted by N.
ZQNR
It is clear that N ⊆ Z ⊆ Q ⊆ R, and this
may be represented by this diagram.
The set of all x such that (. . .) is denoted
by {x:(...)}.
Thus {x:0< x < 1} is the set of all real numbers between 0 and 1.
{x: x > 0, x rational} is the set of all positive rational numbers.
{2n: n = 0, 1, 2,...} is the set of all even numbers.
Among the most important subsets of R are the intervals. The following is an exhaustive list
of the various types of intervals and the standard notation for them. We suppose that a and b
are real numbers and that a < b:
(a, b) ={x: a < x < b} [a, b] ={x: a ≤ x ≤ b}
(a, b] ={x: a < x ≤ b} [a, b) ={x: a ≤ x < b}
(a, ∞) ={x: a < x} [a, ∞) ={x: a ≤ x}
(−∞, b) ={x: x < b} (−∞, b] ={x: x ≤ b}
Intervals may be represented by diagrams as shown in Example 2.
Example 2
Illustrate each of the following intervals of real numbers:
a [−2, 3] b (−3, 4] c (−∞,5] d (−2, 4) e (−3, ∞)
Solution
6543210–1–2–3–4
6543210–1–2–3–4
6543210–1–2–3–4
6543210
–
1
–2–3–4
6543210–1–2–3–4
a
c
e
b
d
The ‘closed’ circle (•) indicates that the number is included.
The ‘open’ circle (◦) indicates that the number is not included.
The following are subsets of the real numbers for which we have special notations:
R
+
={x: x > 0}
R
−
={x: x < 0}
R \{0} is the set of real numbers excluding 0.
Z
+
={x: x ∈ Z, x > 0}
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FAQs

What are the key characteristics of a function?
A function is defined as a relation where each input from the domain corresponds to exactly one output in the range. This means that for any two ordered pairs (a, b) and (a, c) in the function, b must equal c. To determine if a relation is a function, one can use the vertical line test: if a vertical line intersects the graph at more than one point, the relation is not a function. This concept is fundamental in understanding how functions operate in mathematics.
How do transformations affect the graph of a function?
Transformations such as translations, dilations, and reflections modify the position and shape of the graph of a function. For instance, a translation shifts the graph horizontally or vertically, while a dilation changes its size. A reflection flips the graph over a specified axis. Understanding these transformations allows students to predict how the graph will change based on alterations to the function's equation, which is crucial for graphing and analyzing functions.
What is the significance of one-to-one functions?
One-to-one functions are significant because they ensure that each output is paired with a unique input. This property allows for the existence of an inverse function, which can be used to reverse the mapping of the original function. Identifying one-to-one functions is essential in various mathematical applications, including solving equations and analyzing data. The horizontal line test can be used to determine if a function is one-to-one by checking if any horizontal line intersects the graph more than once.
What types of transformations are discussed in this chapter?
The chapter discusses several types of transformations, including translations, dilations, and reflections. Translations involve shifting the graph along the x-axis or y-axis, while dilations change the size of the graph either vertically or horizontally. Reflections flip the graph over the x-axis or y-axis. Each transformation has specific rules that dictate how the coordinates of points on the graph are altered, which is crucial for accurately sketching transformed functions.

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