**Unformatted text preview: **Unit 3 Functions Introduction Introduction
In mathematics you often work with situations in which one quantity
depends on another. For example:
• The distance walked by a woman at a particular speed depends on the
time that she’s been walking. • The height of a gondola on a Ferris wheel depends on the angle
through which the wheel has rotated since the gondola was in its
lowest position. • The number of 5-litre tins of a particular type of paint needed by a
decorator depends on the area that he intends to paint. Whenever one quantity depends on another, we say that the ﬁrst quantity
is a function of the second quantity. The idea of a function is
fundamental in mathematics, and in particular it forms the foundation for
calculus, which you’ll begin to study in Unit 6.
In this unit you’ll be introduced to the terminology and notation that are
used for functions. You’ll learn about some standard, frequently-arising
types of functions, and how to use graphs to visualise properties of
functions. You’ll also learn how you can use your knowledge about a few
standard functions to help you understand and work with a wide range of
related functions. Later in the unit you’ll revise exponential functions and
logarithms, and practise working with them. In the ﬁnal section you’ll
revise inequalities, and see how working with functions and their graphs
can help you understand and solve some quite complicated inequalities. A Ferris wheel This is a long unit. The study calendar allows extra time for you to
study it. 1 Functions and their graphs
This section introduces you to the idea of a function and its graph, and
shows you some standard functions. You’ll start by learning about sets,
which are needed when you work with functions and also in many other
areas of mathematics. 201 Unit 3 Functions 1.1 Sets of real numbers
In mathematics a set is a collection of objects. The objects could be
anything at all: they could be numbers, points in the plane, equations or
anything else. For example, each of the following collections of objects
forms a set:
• all the prime numbers less than 100 • all the points on any particular line in the plane • all the equations that represent vertical lines • the solutions of any particular quadratic equation. A set can contain any number of objects. It could contain one object, two
objects, twenty objects, inﬁnitely many objects, or even no objects at all.
Each object in a set is called an element or member of the set, and we
say that the elements of the set belong to or are in the set.
There are many ways to specify a set. If there are just a few elements,
then you can list them, enclosing them in curly brackets. For example, you
can specify a set S as follows:
S = {3, 7, 9, 42}.
Another simple way to specify a set is to describe it. For example, you can
say ‘let T be the set of all even integers’ or ‘let U be the set of all real
numbers greater than 5’. We usually denote sets by capital letters.
The set that contains no elements at all is called the empty set, and is
denoted by the symbol ∅.
It’s often useful to state that a particular object is or is not a member of a
particular set. You can do this concisely using the symbols ∈ and .∈, which
mean ‘is in’ and ‘is not in’, respectively. For example, if S is the set
speciﬁed above, then the following statements are true:
7∈S and Activity 1 10 .∈ S. Understanding set notation Let X = {1, 2, 3, 4} and let Y be the set of all odd integers. Which of the
following statements are true?
(a) 1 ∈ X (b) 1 ∈ Y (c) 2 .∈ X (d) 2 .∈ Y It’s often useful to construct ‘new sets out of old sets’. For example, if A
and B are any two sets, then you can form a new set whose members are
all the objects that belong to both A and B. This set is called the
intersection of A and B, and is denoted by A ∩ B. For instance, if
A = {1, 2, 3, 4} 202 and B = {3, 4, 5}, 1 Functions and their graphs then
A ∩ B = {3, 4}.
Similarly, if A and B are any two sets, then you can form a new set whose
members are all the objects that belong to either A or B (or both). This
set is called the union of A and B, and is denoted by A ∪ B. For example,
if A and B are as speciﬁed above, then
A ∪ B = {1, 2, 3, 4, 5}.
You might ﬁnd it helpful to visualise intersections and unions of sets by
using diagrams like those in Figure 1, which are known as Venn
diagrams. The Venn diagrams in the ﬁgure show the intersection and
union of the particular sets A and B above.
A B
1 3 2 4 (a) A 5 B
1 3 2 4 5 (b) Figure 1 (a) The intersection (shaded) and (b) the union (shaded) of two
sets
Venn diagrams are named after the logician John Venn, who used
them in publications starting in 1880. However, the idea of using
diagrams in this way did not originate with Venn. The proliﬁc Swiss
mathematician Leonhard Euler (pronounced ‘oiler’) used them in his
Letters to a German Princess (1760–62). Venn acknowledged Euler’s
inﬂuence by calling his own diagrams ‘Eulerian circles’. He extended
Euler’s idea, using the diagrams to analyse more complex logical
problems. As well as working on logic at Cambridge University, Venn
was for some time a priest and later a historian. There is more about
Euler on page 214. John Venn (1834–1923)
You can form intersections and unions of more than two sets in a similar
way. In general, the intersection of two or more sets is the set of all
objects that belong to all of the original sets, and the union of two or
more sets is the set of all objects that belong to any of the original sets.
For example, if A and B are as speciﬁed above and
C = {20, 21},
then
A∩B∩C =∅ and A ∪ B ∪ C = {1, 2, 3, 4, 5, 20, 21}. 203 Unit 3 Functions Activity 2 Understanding unions and intersections of sets Let P = {1, 2, 3, 4, 5, 6}, let Q = {2, 4, 6, 8, 10, 12} and let R be the set of
all integers divisible by 3. Specify each of the following sets.
(a) P ∩ Q (b) Q ∩ R (c) P ∩ Q ∩ R (d) P ∪ Q The set membership symbol ∈ is a stylised version of the Greek letter
ε (epsilon). The Italian mathematician Giuseppe Peano (1858–1932),
the founder of symbolic logic, used ε to indicate set membership in a
text published in 1889. He stated that it was an abbreviation for the
Latin word ‘est’, which means ‘is’. The symbol was then adopted by
the logician Bertrand Russell (1872–1970) in a text published in 1903,
but it was typeset in a form that looks like the modern symbol ∈, and
this form has remained in use to the present day. Peano also
introduced the symbols ∩ and ∪ for intersection and union.
The empty set symbol ∅ was introduced in 1939 by the inﬂuential
French mathematician Andr´e Weil (1906–1998). It was inspired by
the letter ø in the Norwegian alphabet. Sometimes every element of a set A is also an element of a set B. In this
case we say that A is a subset of B, and we write A ⊆ B. For example: 1
3 2 • {1, 3} is a subset of {1, 2, 3} (as shown in Figure 2) • the set of integers is a subset of the set of real numbers. Every set is a subset of itself, and the empty set is a subset of every set.
In this module, and particularly in this unit, you’ll mostly be working with
sets whose elements are real numbers. In the rest of this subsection, you’ll
meet some useful ways to visualise and represent sets of this type. Figure 2 A subset of a set
(shaded) The set of all real numbers is denoted by R. You can handwrite this as:
You saw in Unit 1 that you can visualise the real numbers as points on an
inﬁnitely long straight line, called the number line or the real line. Part
of the number line is shown in Figure 3. Although only the integers are
marked in the diagram, every point on the line represents a real number.
− 10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 9 10 Figure 3 The number line 204 1 Functions and their graphs You can use the number line to visualise sets of real numbers. For example:
• Figure 4(a) shows the set {−1, 0, 1}. • Figure 4(b) shows the set of real numbers that are greater than or
equal to 2 and also less than or equal to 6. • Figure 4(c) shows the set of real numbers that are greater than −5. • Figure 4(d) shows the set of real numbers that are less than
greater than or equal to 3. 1
2 or In these kinds of diagrams, a solid dot indicates a number that’s included
in the set, and a hollow dot indicates a number that isn’t included. A
heavy line that continues to the left or right end of the diagram indicates
that the set extends indeﬁnitely in that direction. −3−2−1 0 1 2 3
(a) −7−6−5−4−3−2−1 0 1 2 3 4 5 6 7 8
(b) −3−2−1 0 1 2 3 4 5
1
2 (c) (d) Figure 4 Sets of real numbers
The sets in Figure 4(b) and (c) are examples of a special type of set of real
numbers, called an interval. An interval is a set of real numbers that
corresponds to a part of the number line that you can draw ‘without lifting
your pen from the paper’. The sets in Figure 4(a) and (d) aren’t intervals,
as they have ‘gaps’ in them. In fact, the set in Figure 4(a) is the union of
three intervals (each containing a single number), and the set in
Figure 4(d) is the union of two intervals.
A number that lies at an end of an interval is called an endpoint of the
interval. For example, the interval in Figure 4(b) has two endpoints,
namely 2 and 6, and the interval in Figure 4(c) has one endpoint,
namely −5. The whole set of real numbers, R, is an interval with no
endpoints.
An interval that includes all of its endpoints is said to be closed, and one
that doesn’t include any of its endpoints is said to be open. For example,
the interval in Figure 4(b) is closed (since it includes both its endpoints),
and the one in Figure 4(c) is open (since it excludes its single endpoint).
An interval that includes one endpoint and excludes another, such as the
interval in Figure 5, is said to be half-open (or half-closed). Since the
interval R has no endpoints, it’s both open and closed! This fact may seem
strange at the moment, but it will make more sense if you go on to study
pure mathematics at higher levels. −2−1 0 1 2 3 4 Figure 5 A half-open (or
half-closed) interval 205 Unit 3 Functions Activity 3 Recognising intervals State whether each of the sets below is an interval. For each set that is an
interval, state whether it’s open, closed or half-open.
(a) −3−2−1 0 1 2 3
(c) −3−2−1 0 1 2 3
(e) −3−2−1 0 1 2 3
(g) −3−2−1 0 1 2 3 (b) −3−2−1 0 1 2 3
(d) −3−2−1 0 1 2 3
(f) −3−2−1 0 1 2 3
(h) −3−2−1 0 1 2 3 A convenient way to describe most intervals is to use inequality signs.
These are listed below, with their meanings. (Note that some texts use
slightly diﬀerent inequality signs: ! and " instead of ≤ and ≥.) Inequality signs
<
≤
>
≥ is
is
is
is less than
less than or equal to
greater than
greater than or equal to For example, the interval in Figure 6(a) is the set of real numbers x such
that x > 2 (that is, such that x is greater than 2).
Similarly, the interval in Figure 6(b) is the set of real numbers x such that
x > 1 and x ≤ 4 (that is, such that x is greater than 1 and x is less than or
equal to 4). We usually write this description slightly more concisely, as
follows: the interval is the set of real numbers x such that 1 < x ≤ 4 (that
is, such that 1 is less than x, which is less than or equal to 4). 206 1 0 1 2 3 4 5 6 7
(a) Functions and their graphs −1 0 1 2 3 4 5 6
(b) Figure 6 Intervals
It might help you to remember the meanings of the inequality signs if you
notice that when you use either of the signs < or >, the lesser quantity is
on the smaller, pointed side of the sign. The same is true for the signs ≤
and ≥, except that one quantity is less than or equal to the other, rather
than deﬁnitely less than it.
The statement ‘x > 2’ is called an inequality. In general, an inequality is
a mathematical statement that consists of two expressions with an
inequality sign between them. A statement such as ‘1 < x ≤ 4’ is called a
double inequality. The two inequality signs in a double inequality
always point in the same direction as each other. Activity 4 Using inequality signs to describe intervals (a) Draw diagrams similar to those in Figure 6 to illustrate the intervals
described by the following inequalities and double inequalities.
(i) 0 < x < 1 (ii) −3 ≤ x < 2 (iii) x ≤ 5 (iv) x > 4 (b) For each of the following diagrams, write down an inequality or double
inequality that describes the interval illustrated.
(i) −3−2−1 0 1 2 3 4 5 6
(iii) −4−3−2−1 0 1
(v) −1 0 1 2 3 4 5 6 7 (ii) −5−4−3−2−1 0
(iv) −5−4−3−2−1 0 1
(vi) 2 3 4 5 6 7 8 Another useful way to describe intervals is to use interval notation. For
example, the interval described by the double inequality 4 ≤ x < 7 is
denoted in interval notation by [4, 7). The square bracket indicates an
included endpoint, and the round bracket indicates an excluded one. An
interval that extends indeﬁnitely is denoted by using the symbol ∞ (which
is read as ‘inﬁnity’), or its ‘negative’, −∞ (which is read as ‘minus
inﬁnity’), in place of an endpoint. For example, the interval described by
the inequality x ≥ 5 is denoted by [5, ∞), and the interval described by the 207 Unit 3 Functions
inequality x < 6 is denoted by (−∞, 6). We always use a round bracket
next to ∞ or −∞ in interval notation. Here’s a summary of the notation. Interval notation
Open intervals
(a, b)
❝ a ❝ a<x<b b ❝ a Closed intervals
[a, b]
! a ! a≤x≤b b ! a (a, ∞) (−∞, b) x>a x<b [a, ∞) (−∞, b] x≥a x≤b (−∞, ∞) ❝ b R ! b (−∞, ∞) {a} R x=a ! Half-open (or half-closed) intervals
[a, b)
(a, b]
! a ❝ a≤x<b b ❝ a ! a<x≤b b Notice that you’ve now seen two diﬀerent meanings for the notation (a, b),
where a and b are real numbers. It can mean either an open interval, or a
point in the coordinate plane. The meaning is usually clear from the
context. Activity 5 Using interval notation Write each of the intervals below in interval notation.
(a) −3−2−1 0 1 2 3 4 5 6
(c) −4−3−2−1 0 1
(e) −1 0 1 2 3 4 5 6 7 (b) −5−4−3−2−1 0
(d) −5−4−3−2−1 0 1
(f) 2 3 4 5 6 7 8 Sometimes you need to work with sets of real numbers that are unions of
intervals, like those in Figure 7. 208 1 −2−1 0 1 2 3 4 5
(a) Functions and their graphs −1 0 1 2 3 4 5 6 7 8 9
(b) Figure 7 Two unions of intervals
You can denote a union of intervals in interval notation by using the usual
notation for intervals together with the union symbol ∪. For example, the
sets in Figure 7 can be written as
(−∞, 1] ∪ [2, 4) and [0, 3) ∪ (3, 5] ∪ [7, 8], respectively. Activity 6 Denoting unions of intervals For each of the following diagrams, write the set illustrated in interval
notation. (a) (b) −8−7−6−5−4−3−2−1 0 1 2 3 0 1 2 3 4 5 6 7 (c) −3−2−1 0 1 2 3 It’s often useful to state that a particular number lies in, or doesn’t lie in, a
particular interval or union of intervals. You can do this concisely using the
symbols ∈ and .∈ in the usual way. For example, as illustrated in Figure 8,
1 ∈ [0, 4] and − 1 .∈ [0, 4]. In the next subsection you’ll begin your study of functions. −1 0 1 2 3 4 5 Figure 8 The interval [0, 4] 1.2 What is a function?
As mentioned in the introduction to this unit, whenever one quantity
depends on another, we say that the ﬁrst quantity is a function of the
second quantity. Here are some more examples.
• If a car is driving along a straight road, then its displacement s (in
km) from some reference point depends on the time t (in hours) that
has elapsed since the start of its journey. So s is a function of t. 209 Unit 3 Functions
• The formula
C = 2πr
expresses the circumference C of a circle in terms of its radius r (with
both C and r measured in the same units). So the value of C depends
on the value of r, and hence C is a function of r. • Figure 9
An electrocardiogram
(each high peak in voltage
corresponds to a heartbeat) The electrical voltage between two points on a person’s skin either
side of his or her heart (which can be measured using electrodes)
changes rhythmically with every heartbeat. So the voltage V (in volts,
say) depends on the time t (in seconds, say) that has elapsed since
some point in time, and hence V is a function of t. There’s no simple
formula for the relationship between t and V , but it’s often displayed
as an electrocardiogram (ECG), like the one in Figure 9. In each of these examples, there’s a rule that converts each value of one
variable (such as t, in the car example) to a value of the other variable
(such as s, in the car example). You can think of the rule as a kind of
processor that takes input values and produces output values, as
illustrated in Figure 10.
input value processor output value Figure 10 A processor that takes input values and produces output
values
For instance, in the car example, an input value of 1.2 (a time, in hours)
might be converted by the processor to an output value of 60 (a
displacement, in kilometres). Similarly, in the circle example, an input
value of 3 (a radius, in centimetres) would be converted by the processor to
an output value of 2π × 3 = 6π (a circumference, in centimetres).
Sometimes the rule associated with a function can be expressed using a
formula, and sometimes it can’t.
In each of the three examples in the list above, there’s also a set of allowed
input values, and a set of values within which every output value lies. For
instance, with the car example, if the journey lasts three hours, then the
allowed input values are the real numbers between 0 and 3 inclusive (the
possible elapsed times, in hours), and the output values lie in the set R of
real numbers (they are displacements of the car from the reference point,
in kilometres).
A function is a mathematical object that describes a situation like those
listed above. It’s deﬁned as follows. 210 1 Functions and their graphs A function consists of:
• a set of allowed input values, called the domain of the function • a set of values in which every output value lies, called the
codomain of the function • a process, called the rule of the function, for converting each
input value into exactly one output value. It’s often useful to denote a function by a letter. If a function is denoted
by f , say, then for any input value x, the corresponding output value is
denoted by f (x), which is read as ‘f of x’.
For example, suppose that we denote the function associated with the car
example by f . If the rule of this function converts the input value 1.2 to
the output value 60, then we write
f (1.2) = 60.
Similarly, suppose that we denote the function associated with the circle
example by g. The rule of this function converts the input value 3 to the
output value 6π, so we write
g(3) = 6π.
This type of notation is known as function notation.
One use of function notation is for specifying the rule of a function, when
this can be done using a formula. For example, suppose that h is the
function whose domain and codomain each consist of all the real numbers,
and whose rule is ‘square the input number’. Then, for example,
h(2) = 4, h(5) = 25 and h(−1) = 1, and the rule of h can be written as
h(x) = x2 .
Similarly, the rule of the function associated with the circle example can
be written as g(r) = 2πr.
When you write down the rule of a function, it doesn’t matter what letter
you use to represent the input value. So the rule of the function h above
could also be written as, for example,
h(t) = t2 or h(u) = u2 . The variable used to denote the input value of a function is sometimes
called the input variable.
It’s traditional to use the letters f , g and h for functions, and the letters
x, t and u for input variables. Although you can use any letters, these ones
are often used in general discussions about functions. The most standard
letters are f for a function and x for an input variable. 211 Unit 3 Functions Activity 7 Understanding function notation (a) Suppose that f is the function whose domain and codomain each
consist of all the real numbers, and whose rule is f (t) = 4t. Write
down the values of f (5) and f (−3).
(b) Suppose that g is the function whose domain and codomain each
consist of all the real numbers, and whose rule can be written in words
as ‘multiply the input...

View
Full Document