Functions 2

2
Functions
Assessment statements
2.1
Concept of a function f: x → f(x); domain, range, image (value).
Composite functions f  g; identity function. Inverse function f21.
2.2
The graph of a function; its equation y 5 f(x).
Function graphing skills.
Investigation of key features of graphs.
Solutions of equations graphically.
2.3
Transformations of graphs: translations, stretches, reflections in the
axes.
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The graph of y 5 f21(x) as the reflection in the line y 5 x of the graph
y 5 f(x).
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1 from y 5 f(x).
The graph of y 5 ___
f(x)
The graphs of the absolute value functions, y 5 |f(x)| and y 5 f(|x|).
1 , x 0: its graph; its self-inverse nature.
The reciprocal function x → __
x
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2.4
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Introduction
The relationship between two quantities – how the value of one quantity
depends on the value of another quantity – is the key behind the concept
of a function. Functions and how we use them are at the very foundation
of many topics in mathematics, and are essential to our understanding
of much of what will be covered later in this book. This chapter will
look at some general characteristics and properties of functions. We will
consider composite and inverse functions, and investigate how the graphs
of functions can be transformed by means of translations, stretches and
reflections.
L
θ
Figure 2.1 A simple pendulum.
46
2.1
Definition of a function
A simple pendulum consists of a heavy object hanging from a string of
length L (in metres) and fixed at a pivot point (Figure 2.1). If you displace
the suspended object to one side by a certain angle  from the vertical and
release it, the object will swing back and forth under the force of gravity.
The period T (in seconds) of the pendulum is the time for the object to
return to the point of release and, for a small
angle , the two variables T
__
L where g is the gravitational
and L are related by the formula T 5 2p __
g
√
field strength (acceleration due to gravity). Therefore, assuming that the
force of gravity is constant at a given elevation (g  9.81 m s22 at sea level),
the formula can be used to calculate the value of T for any value of L.
As with the period T and the length L for a pendulum, many mathematical
relationships concern how the value of one variable determines the value
of a second variable. Other examples include:
Converting degrees Celsius to
degrees Fahrenheit:
F 5 _95C 1 32
r
°C
50
40
30
20
pa
g
A
°F
120
110
100
90
80
70
60
50
40
30
20
10
0
es
Area of a circle determined
by its radius:
A 5 pr 2 (p is a constant)
10
0
�10
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�20
x
| x | units
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| x | units
m
Distance that a number is from the origin determined by its absolute value:
0
0
x
In general, suppose that the values of a particular independent variable,
for example x, determine the values of a dependent variable y in such a
way that for a specific value of x, a single value of y is determined. Then we
say that y is a function of x and we write y 5 f (x) (read ‘y equals f of x’), or
y 5 g(x) etc., where the letters f and g represent the name of the function.
For the four mathematical relationships that were described above, we have:
__
√
Period T is a function of length L: T 5 2p
Area A is a function of radius r : A 5
__
√
L , or f (L) 5 2p __
L where T 5 f (L).
__
g
g
pr 2, or
g(r) 5
pr 2
where A 5 g(r).
°F (degrees Fahrenheit) is a function of °C : F 5 _95C 1 32, or
t(C) 5 _95 C 1 32 where F 5 t(C).
Distance y from origin is a function of x: y 5 |x|, or f (x) 5 |x| where y 5 f (x).
Along with equations, other useful ways of representing a function include
a graph of the equation on a Cartesian coordinate system (also called
47
2
Functions
a rectangular coordinate system), a table, a set of ordered pairs, or a
mapping. These are illustrated below for the absolute value function y 5 |x|.
Table y 5 |x|
Graph
y
10
y�|x|
5
René Descartes
�10
�5
0
5
210
10
15
2 __
15
__
25
5
23.6
3.6
2
2
x
10
y
0
0
__
�5
√2
5
5
8.3
8.3
pa
g
es
Hint: The coordinate system for the graph
of an equation has the independent variable
on the horizontal axis and the dependent
variable on the vertical axis.
__
√2
Set of ordered pairs
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The graph of the equation y 5 |x|
consists of an infinite set of ordered
pairs (x, y) such that each is a
solution of the equation. The
following set includes some of the
ordered pairs on the line:
__
__
m
{(223, 23), (210, 10), (2√7 , √7 ),
(0, 0), (5, 5)}.
10
10
Mapping
10
�10
�5
5
�3.6
0
3.6
5
10
0
y�|x|
The largest possible set of values for the independent variable (the input
set) is called the domain – and the set of resulting values for the dependent
variable (the output set) is called the range. In the context of a mapping,
each value in the domain is mapped to its image in the range.
Sa
The Cartesian coordinate
system is named in honour of
the French mathematician and
philosopher René Descartes
(1596–1650). Descartes
stimulated a revolution in
the study of mathematics by
merging its two major fields
– algebra and geometry. With
his coordinate system utilizing
ordered pairs (Cartesian
coordinates) of real numbers,
geometric concepts could
be formulated analytically
and algebraic concepts (e.g.
relationships between two
variables) could be viewed
graphically. Descartes initiated
something that is very helpful
to all students of mathematics
– that is, considering
mathematical concepts
from multiple perspectives:
graphical (visual) and analytical
(algebraic).
x
All of the various ways of representing a mathematical function illustrate
that its defining characteristic is that it is a rule by which each number in
the domain determines a unique number in the range.
Definition of a function
A function is a correspondence (mapping) between two sets X and Y in which each
element of set X corresponds to (maps to) exactly one element of set Y. The domain is
set X (independent variable) and the range is set Y (dependent variable).
Not all equations represent a function. The solution set for the equation
x 2 1 y 2 5 1 is the set of ordered pairs (x, y) on the circle of radius equal
to 1 and centre at______
the origin (see Figure 2.2). If we solve the equation for
y, we get y 5 6√ 1 2 x 2 . It is clear that any value of x between 21 and 1
will produce two different values of y (opposites). Since at least one value
in the domain (x) determines more than one value in the range (y), then
48
the equation does not represent a function. A correspondence between two
sets that does not satisfy the definition of a function is called a relation.
y
1
y1
Alternative definition of a function
A function is a relation in which no two different ordered pairs have the same first
coordinate.
x
�1
1
x
y2
A vertical line intersects the graph of a function at no more than one point (vertical
line test).
�1
y
y
Figure 2.2 Graph of x 2 1 y 2 5 1.
x
es
x
At least one vertical line intersects the
graph at more than one point, so y is
not a function of x.
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Any vertical line intersects the graph at
no more than one point, so y is a
function of x.
m
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Not only are functions important in the study of mathematics and
science, we encounter and use them routinely – often in the form of
tables. Examples include height and weight charts, income tax tables, loan
payment schedules, and time and temperature charts. The importance
of functions in mathematics is evident from the many functions that are
installed on your GDC.
SIN
x21
Sa
For example, the keys labelled
LN
_
√
each represent a function, because for each
input (entry) there is only one output (answer).
The calculator screen image shows that for the
function y 5 1n x, the input of x 5 10 has only
one output of y  2.302 585 093.
ln(10)
2.302585093
For many physical phenomena, we observe that one quantity depends
on another. The word function is used to describe this dependence of
one quantity on another – i.e. how the value of an independent variable
determines the value of a dependent variable. A common mathematical
task is to find how to express one variable as a function of another variable.
Example 1
a) Express the volume V of a cube as a function of
the length e of each edge.
e
b) Express the volume V of a cube as a function of
its surface area S.
e
e
49
2
Functions
Solution
a) V as a function of e is V 5 e 3.
b) The surface area of the cube consists of six squares each with an area of
e 2. Hence, the surface area is 6e 2; that is, S 5 6e 2. We need to write V in
terms of S. We can do this by first expressing e in terms of S, and then
substituting this expression in
for e in the equation V 5 e 3.
__
S ⇒ e 5 __
S.
S 5 6e 2 ⇒ e 2 5 __
6
6
Substituting,
√
__
V5
_1
__
(S )
S 5 _____
S
S  S 5 __
S __
5 __
(√  __6S ) 5 _____
6 √6
3
3
_3
(62)3
62
2
_1
1
2
_3
_1
2
_1
61  6 2
__
√
S __
S.
V as a function of S is V 5 __
es
6 6
Example 2 – Finding a function in terms of a single variable
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An offshore wind turbine is located at point W, 4 km offshore from the
nearest point P on a straight coastline. A maintenance station is at point
M, 3 km down the coast from P. An engineer is returning by boat from
the wind turbine. He decides to row to a dock at point D that is located
between P and M at an unknown distance x km from point P. The engineer
can row 3 km/hr and walk 6 km/hr. Express the total time
T (hours) for the trip from the wind turbine to the maintenance station as
a function of x (km).
W
4
P
D
M
x
3
Solution
distance.
To get an equation for T in terms of x, we use the fact that time 5 _______
rate
We then have
distance DM
distance WD 1 ___________
T 5 ___________
3
6
The distance WD can be expressed in terms of x by using Pythagoras’
theorem.
_______
WD 2 5 x 2 1 42 ⇒ WD 5 √x 2 1 16
To express T in terms of only the single variable x, we note that DM 5 3 2 x.
50
Then the total time T can be written in terms of x by the equation:
_______
T5
√ x 2 1 16
________
3
_______
3 2 x or T 5 __
1√ x 2 + 16 1 __
1 2 __x
1 _____
3
2 6
6
Using our graphic display calculator (GDC) to graph the equation gives
a helpful picture showing how T changes when x changes. In function
graphing mode on a GDC, the independent variable is always x and the
dependent variable is always y.
Zooming in on the graph indicates that
there is a value for x between 1.5 and
3 that will make the time for the trip a
minimum. In Chapter 13, we will use
calculus techniques to find the value of x
that gives a minimum time for the trip.
WINDOW
Xmin=1.5
Xmax=3
Xscl=1
Ymin=1.64
Ymax=1.68
Yscl=1
Xres=1
es
Y2�
Y3�
Y4�
Y5�
Y6�
WINDOW
Xmin=0
Xmax=3
Xscl=1
Ymin=0
Ymax=2
Yscl=1
Xres=1
pa
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Plot1 Plot2 Plot3
Y1 �(1 3) ( X2�16
(�1 2–X 6
Domain and range of a function
Sa
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The domain of a function may be stated explicitly, or it may be implied by
the expression that defines the function. Except in Chapter 10, where we
will encounter functions for which the variables can have values that are
imaginary numbers, we can assume that any functions that we will work
with are real-valued functions of a real variable. That is, the domain and
range will only contain real numbers or some subset of the real numbers.
Therefore, if not explicitly stated otherwise, the domain of a function is
the set of all real numbers for which the expression is defined as a real
number. For example, if a certain value of x is substituted into the algebraic
expression defining a function and it causes division by zero or the square
root of a negative number (both undefined in the real numbers) to
occur, that value of x cannot be in the domain. The domain of a function
may also be implied by the physical context or limitations that exist in a
problem.__For example, for both functions derived in Example 1
S __
S and V 5 e 3 the domain is the set of positive real numbers
V 5 __
6 6
(symbolized by R1) because neither a length (edge of a cube) nor a surface
area (face of a cube) can have a value that is negative or zero. In Example 2
the domain for the function is 0 , x , 3 because of the constraints given
in the problem. Usually the range of a function is not given explicitly and
is determined by analyzing the output of the function for all values of the
input (domain). The range of a function is often more difficult to find
than the domain, and analyzing the graph of a function is very helpful in
determining it. A combination of algebraic and graphical analysis is very
useful in determining the domain and range of a function.
( 
√
)
51
2
Functions
Example 3 – Domain of a function
Find the domain of each of the following functions.
a) {(26, 23), (21, 0), (2, 3), (3, 0), (5, 4)}
b) Volume of a sphere: V 5 _43pr 3
5
c) y 5 ______
2x 2 6
_____
d) y 5 √ 3 2 x
Solution
a) The function consists of a set of ordered pairs. The domain of the
function consists of all first coordinates of the ordered pairs. Therefore,
the domain is the set x  {26, 21, 2, 3, 5}.
b) The physical context tells you that a sphere cannot have a radius that is
negative or zero. Therefore, the domain is the set of all real numbers r
such that r . 0.
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c) Since division by zero is not defined for real numbers then 2x 2 6  0.
Therefore, the domain is the set of all real numbers x such that
x  R, x  3.
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d) Since the square root of a negative number is not real, then 3 2 x  0.
Therefore, the domain is all real numbers x such that x  3.
Example 4 – Domain and range of a function I
Solution
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8
6
2
1
2
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range
4
3 x
domain
Figure 2.3 The graph of y = x 2.
Table 2.1 Different ways of
expressing the domain and range
of y 5 x 2.
52
• Algebraic analysis: Squaring any real number produces another real
number. Therefore, the domain of y 5 x 2 is the set of all real numbers
(R). What about the range? Since the square of any positive or negative
number will be positive and the square of zero is zero, the range is the
set of all real numbers greater than or equal to zero.
m
y
10
�3 �2 �1
�2
e
What is the domain and range for the function y 5 x 2?
• Graphical analysis: For the domain, focus on the x-axis and horizontally
scan the graph from 2 to 1. There are no ‘gaps’ or blank regions
in the graph and the parabola will continue to get ‘wider’ as x goes to
either 2 or 1. Therefore, the domain is all real numbers. For the
range, focus on the y-axis and vertically scan from 2 or 1. The
parabola will continue ‘higher’ as y goes to 1, but the graph does not
go below the x-axis. The parabola has no points with negative
y-coordinates. Therefore, the range is the set of real numbers greater
than or equal to zero. See Figure 2.3.
Description in words
Interval notation (both formats)
domain is any real number
domain is {x : x  R}, or domain is x  ]2, [
range is any real number
greater than or equal to zero
range is {y : y  0}, or range is y  [0, [
Function notation
It is common practice to name a function using a single letter, with f, g
and h being the most common. Given that the domain variable is x and
the range variable is y, the symbol f (x) denotes the unique value of y that
is generated by the value of x. Another notation – sometimes referred to
as mapping notation – is based on the idea that the function f is the rule
that maps x to f (x) and is written f : x ↦ f (x). For each value of x in the
domain, the corresponding unique value of y in the range is called the
function value at x, or the image of x under f. The image of x may be
written as f (x) or as y. For example, for the function f (x) 5 x 2: ‘f (3) 5 9’;
or ‘if x 5 3 then y 5 9’.
Notation
Table 2.2 Function notation.
Description in words
‘the function f, in terms of x, is x 2’; or, simply, ‘f of x equals x 2’
es
f (x) 5 x 2
Hint: When asked to determine
the domain and range of a function,
it is wise for you to conduct
both algebraic and graphical
analysis – and not rely too much
on either approach. For graphical
analysis of a function, producing a
comprehensive graph on your GDC is
essential, i.e. a graph that shows all
important features of the graph.
Hint: It is common to write
y 5 f (x) and call it a function but
‘the function f maps x to x 2’
f (3) 5 9
‘the value of the function f when x 5 3 is 9’; or, simply, ‘f of 3 equals 9’
f:3 ↦ 9
‘the image of 3 under the function f is 9’
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f : x ↦ x2
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Example 5 – Domain and range of a function II
this can be considered a misuse of
the notation. If we were to be very
precise, we would call f the function
and f (x) the value of the function at
x. But this is often overlooked and
we accept writing expressions such
as y 5 x 2 or y 5 sin x and calling
them functions.
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1 .
Find the domain and range of the function h : x ↦ _____
x22
m
Solution
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• Algebraic analysis: The function produces a real number for all x, except
for x 5 2 when division by zero occurs. Hence, x 5 2 is the only real
1 can never be
number not in the domain. Since the numerator of _____
x22
zero, the value of y cannot be zero. Hence, y 5 0 is the only real number
not in the range.
• Graphical analysis: A horizontal scan shows a ‘gap’ at x 5 2 dividing the
graph of the equation into two branches that both continue indefinitely,
with no other ‘gaps’ as x → 6 . Both branches are asymptotic
(approach but do not intersect) to the vertical line x 5 2. This line is a
vertical asymptote and is drawn as a dashed line (it is not part of the
graph of the equation). A vertical scan reveals a ‘gap’ at y 5 0 (x-axis)
with both branches of the graph continuing indefinitely, with no other
‘gaps’ as y → 6 . Both branches are also asymptotic to the x-axis.
The x-axis is a horizontal asymptote.
1 :
Both approaches confirm the following for h : x ↦ _____
x22
1
h(x) � x � 2
y
4
2
2
�2
4 x
�2
�4
The domain is {x : x  R, x  2} or x  ]2, 2[  ]2, [
The range is
{y : y  R, y  0} or y  ]2, 0[  ]0, [
53
2
Functions
Example 6 – Domain and range of function II
_____
Consider the function g (x) 5 √x 1 4 .
a) Find: (i) g (7)
(ii) g (32)
(iii) g (24)
y
3
g(x) � x � 4
b) Find the values of x for which g is undefined.
2
c) State the domain and range of g.
1
�4
Solution
2
�2
4
�1
x
a)
_____
___
(i) g (7) 5 √7______
1 4 5 √11___
 3.32 (3 significant figures)
√ 36 5 6
1
4
5
(ii) g (32) 5 √32
__
_______
(iii) g (24) 5 √24 1 4 5 √0 5 0
b) g (x) will be undefined (square root of a negative) when x 1 4 , 0.
x 1 4 , 0 ⇒ x , 24. Therefore, g (x) is undefined when x , 24.
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c) It follows from__ the result in b) that the domain of g is {x : x  24}.
The symbol √ stands for the principal square root that, by definition,
can only give a result that is positive or zero. Therefore, the range of g is
{y : y  0}. The domain and range are confirmed by analyzing the graph
of the function.
Example 7 – Domain and range of a function III
Find the domain and range of the function
1 .
______
f (x) 5 _______
√9 2 x 2
X=0
Y=.33333333
m
Solution
1
______
The graph of y 5 _______
on a GDC, shown above, agrees with algebraic
√9 2 x 2
1
______
analysis indicating that the expression _______
will be positive for all x,
√9 2 x 2
and is defined only for 23 , x , 3.
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Hint: As Example 7 illustrates,
it is dangerous to completely
trust graphs produced on a GDC
without also doing some algebraic
thinking. It is important to mentally
check that the graph shown is
comprehensive (shows all important
features of the graph), and that the
graph agrees with algebraic analysis
of the function – e.g. where should
the function be zero, positive,
negative, undefined, increasing/
decreasing without bound, etc.
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Y1=1/ √(9-X2)
y
3
y�
1
9 � x2
4
x
2
1
�4
2
�2
�1
Further analysis and tracing the graph reveals that f (x) has a minimum at
( 0, _13 ). The graph on the GDC (next page) is misleading in that it appears
to show that the function has a maximum value (y) of approximately
2.803 7849. Can this be correct? A lack of algebraic thinking and overreliance on your GDC could easily lead to a mistake. The graph abruptly
stops its curve upwards because of low screen resolution.
54
Function values should get quite large for
values of x a
______
√
little less than 3, because the value of 9 2 x 2 will be
1
______
small, making the fraction _______
large. Using your
√9 2 x 2
GDC to make a table for f (x), or evaluating the function
for values of x very close to 23 or 3, confirms that as x
approaches 23 or 3, y increases without bound, i.e. y goes to
1. Hence, f (x) has vertical asymptotes of x 5 23 and
x 5 3. This combination of graphical and algebraic
analysis leads to the conclusion that the domain of f (x) is
{x : 23 , x , 3}, and the range of f (x) is {y : y  _13}.
Y1=1/ √(9-X2)
TABLE SETUP
TblStart=2.999
Tbl=.0001
Indpnt: Auto Ask
Depend: Auto Ask
X=2.9787234 Y=2.8037849
X
Y1
2.9994
2.9995
2.9996
2.9997
2.9998
2.9999
3
16.668
18.258
20.413
23.571
28.868
40.825
ERROR
Y 1(2.99999)
129.0995525
Y 1(2.999999)
408.2483245
Y 1(2.9999999)
1290.994449
X=2.9994
Exercise 2.1
5 y522 x
2
8 y 5 __
7 y3 5 x
y
4
B
�4 �2
�2
E
2
4 x
�4 �2
�2
�4
y
4
y
4
H
2
4x
2
�4 �2
�2
K
4
2
4x
4 x
�4
2
4x
2
4x
2
4x
�4
y
4
�4 �2
�2
y
4
I
2
2
4x
�4 �2
�2
�4
y
4
�4 �2
�2
4x
�4
L
y
4
2
2
2
2
2
�4
y
�4
F
2
�4
�4 �2
�2
�4 �2
�2
2
2
J
4x
y
4
�4
�4 �2
�2
2
e
�4 �2
�2
�4
�4
y
4
2
G
4x
pl
2
y
4
C
2
Sa
D
y
4
2
2
�4 �2
�2
9 x2 1 y 5 2
x
m
A
6 y 5 x2 1 2
pa
g
4 x2 1 y2 5 4
es
For each equation 1–9, a) match it with its graph (choices are labelled A to L), and
b) state whether or not the equation represents a function – with a justification.
Assume that x is the independent variable and y is the dependent variable.
2 y 5 23
3 x2y5 2
1 y 5 2x
2
4x
�4 �2
�2
�4
55
2
Functions
10 Express the area, A, of a circle as a function of its circumference, C.
11 Express the area, A, of an equilateral triangle as a function of the length, ,, of
each of its sides.
12 A rectangular swimming pool with dimensions 12 metres by 18 metres is
surrounded by a pavement of uniform width x metres. Find the area of the
pavement, A, as a function of x.
13 In a right isosceles triangle, the two equal sides have length x units and the
hypotenuse has length h units. Write h as a function of x.
es
14 The pressure P (measured in kilopascals, kPa) for a particular sample of gas is
directly proportional to the temperature T (measured in kelvin, K) and inversely
proportional to the volume V (measured in litres, ,). With k representing the
constant of proportionality, this relationship can be written in the form of
the equation P 5 k__T .
V
a) Find the constant of proportionality, k, if 150 , of gas exerts a pressure of
23.5 kPa at a temperature of 375 K.
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b) Using the value of k from part a) and assuming that the temperature is held
constant at 375 K, write the volume V as a function of pressure P for this
sample of gas.
15 In physics, Hooke’s law states that the force F (measured in newtons, N) needed
to keep a spring stretched a displacement of x units beyond its natural length is
directly proportional to the displacement x. Label the constant of proportionality
k (known as the spring constant for a particular spring).
a) Write F as a function of x.
pl
e
b) If a spring has a natural length of 12 cm and a force of 25 N is needed to keep
the spring stretched to a length of 16 cm, find the spring constant k.
c) What force is needed to keep the spring stretched to a length of 18 cm?
m
In questions 16–23, find the domain of the function.
16 {(26.2, 27), (21.5, 22), (0.7, 0), (3.2, 3), (3.8, 3)}
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17 Surface area of a sphere: S 5 4p r 2
18 f (x) 5 _25x 2 7
19 h : x ↦ x 2 2 4
20 g (t) 5 √ 3 2 t
21 h(t) 5 √t
_____
3 _
______
√
6
1 21
23 f(x) 5 __
22 f : x ↦ ______
x2 2 9
x2
24 Do all linear equations represent a function? Explain.
_____
25 Consider the function h(x) 5 √ x 2 4 .
a) Find: (i) h(21)
(ii) h(53)
(iii) h(4)
b) Find the values of x for which h is undefined.
c) State the domain and range of h.
In questions 26–30, a) find the domain and range of the function, and b) sketch a
comprehensive graph of the function clearly indicating any intercepts or asymptotes.
1
26 f : x ↦ _____
x25
1
______
27 g(x) 5 _______
√ x2 2 9
2x 2 1
28 h(x) 5 ______
x12
1
__
30 f(x) 5 2 4
29 p : x ↦ √5 2 2x 2
x
56
_______
2
Functions
Similarly to part a) we can see a change from the graph of a function
to the graph of the function of the absolute value. Any portion of
the graph of g(x) or h(x) that was left of the y-axis is eliminated, and
any portion that was to the right of the y-axis is reflected to the left of
the y-axis. Since the portion that was right of the y-axis remains, the
resulting graph is always symmetric about the y-axis.
Summary of transformations on the graphs of functions
Assume that a, h and k are positive real numbers.
Transformed
function
y 5 f (x) 1 k
y 5 f (x) 2 k
y 5 f (x 2 h)
y 5 f (x 1 h)
y 5 2f (x)
y 5 f (2x)
y 5 af (x)
y 5 f (ax)
y 5 |f (x)|
y 5 f (|x|)
Transformation performed on y 5 f (x)
e
pa
g
es
vertical translation k units up
vertical translation k units down
horizontal translation h units right
horizontal translation h units left
reflection in the x-axis
reflection in the y-axis
vertical stretch (a . 1) or shrink (0 , a , 1)
horizontal stretch (0 , a , 1) or shrink (a . 1)
portion of graph of y 5 f (x) below x-axis is reflected above x-axis
symmetric about y-axis; portion right of y-axis is reflected over y-axis
pl
Exercise 2.4
In questions 1–14, sketch the graph of f, without a GDC or by plotting points, by
using your knowledge of some of the basic functions shown in Figure 2.17.
m
1 f : x ↦ x2 2 6
2 f : x ↦ (x 2 6) 2
3 f : x ↦ |x | 1 4
_____
5 f : x ↦ 5 1 √x 2 2
1
12
7 f : x ↦ _______
(x 1 5)2
1
6 f : x ↦ _____
x23
8 f : x ↦ 2x3 2 4
9 f : x ↦ 2 |x 2 1| 1 6
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4 f : x ↦ |x 1 4 |
_______
__
10 f : x ↦ √2x 1 3
11 f : x ↦ 3√x
13 f : x ↦ (_12  x )
14 f : x ↦ (2x)3
2
12 f : x ↦ _12x2
In questions 15–18, write the equation for the graph that is shown.
15
16
y
6
y
3
4
2
2
1
�4
2
�2
�2
�4
�6
84
4 x
�8
�6
�4
2 x
�2
�1
17
18 Vertical and horizontal asymptotes shown:
y
1
�4
�2
�1
y
4
2x
2
�2
�3
2
�2
�4
6 x
4
�2
�5
�6
�4
�6
�8
es
y
4
3
2
1
pa
g
19 The graph of f is given. Sketch the
graphs of the following functions.
a) y 5 f (x) 2 3
b) y 5 f (x 2 3)
c) y 5 2f (x)
d) y 5 f (2x)
e) y 5 2f (x)
f ) y 5 f (2x)
g) y 5 2f (x) 1 4
�5 �4 �3 �2 �1 0
�1
1
2
3
4
5 x
�2
e
�3
23 f : x ↦ [3(x 2 1)]2 2 6
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22 p : x ↦ _12(x 1 4)2
21 h : x ↦ 2x 2 1 2
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20 g : x ↦ (x 2 3)2 1 5
pl
In questions 20–23, specify a sequence of transformations to perform on the graph
of y 5 x2 to obtain the graph of the given function.
Without using your GDC, for each function f(x) in questions 24–26 sketch the graph
1 , b) |f (x)| and c) f(|x|). Clearly label any intercepts or asymptotes.
of a) ___
f(x)
24 f (x) 5 _12x 2 4
25 f (x) 5 (x 2 4)(x 1 2)
26 f (x) 5 x 3
Practice questions
_____
1 Let f : x ↦ √ x 2 3 and g : x ↦ x 2 1 2x. The function (f  g)(x ) is defined for all x  R
except for the interval ]a, b [.
a) Calculate the values of a and b.
b) Find the range of f  g.
2 Two functions g and h are defined as g (x ) 5 2x 2 7 and h(x ) 5 3(2 2 x ).
Find: a) g 21(3)
b) (h  g)(6)
85
2
Functions
4 2 x.
3 Consider the functions f (x ) 5 5x 2 2 and g (x ) 5 _____
3
a) Find g 21.
b) Solve the equation (f  g 21)(x ) 5 8.
4 The functions g and h are defined by g : x ↦ x 2 3 and h : x ↦ 2x.
a) Find an expression for (g  h )(x ).
y
b) Show that g 21(14) 1 h 21(14) 5 24.
2
5 The diagram right shows the graph of
y 5 f (x ). It has maximum and minimum points
at (0, 0) and (1, 21), respectively.
a) Copy the diagram and, on the same diagram, �2
draw the graph of y 5 f (x 1 1) 2 _12.
b) What are the coordinates of the minimum
and maximum points of y 5 f (x 1 1) 2 _12?
1
�1
1
2
3x
�1
�2
es
6 The diagram shows parts of the graphs of y 5 x 2 and y 5 2 _12(x 1 5)2 1 3.
� 12 (x
�
5)2
�8
�6
�4
y � x2
2
2
�2
pl
�10
4
�3
e
y�
pa
g
y
6
4 x
m
�2
Sa
The graph of y 5 x 2 may be transformed into the graph of y 5 2 _12 (x 1 5)2 1 3 by
these transformations.
A reflection in the line y 5 0, followed by
a vertical stretch by scale factor k, followed by
a horizontal translation of p units, followed by
a vertical translation of q units.
Write down the value of
a) k
b) p
c) q.
4
_______
, for 24 , x , 4.
7 The function f is defined by f (x ) 5 ________
√ 16 2 x2
a) Without using a GDC, sketch the graph of f.
b) Write down the equation of each vertical asymptote.
c) Write down the range of the function f.
1
8 Let g : x ↦ __
x, x  0.
a) Without using a GDC, sketch the graph of g.
The graph of g is transformed to the graph of h by a translation of 4 units to the left and
2 units down.
b) Find an expression for the function h.
86
c)
(i) Find the x- and y-intercepts of h.
(ii) Write down the equations of the asymptotes of h.
(iii) Sketch the graph of h.
_____
9 Consider f (x ) 5 √ x 1 3 .
a) Find:
(i) f (8)
(ii) f (46)
(iii) f (23)
b) Find the values of x for which f is undefined.
c) Let g : x ↦ x 2 2 5. Find (g  f )(x ).
x28
10 Let g (x ) 5 _____ and h (x ) 5 x 2 2 1.
2
a) Find g 21(22).
b) Find an expression for (g 21  h )(x ).
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c) Solve (g 21  h )(x ) 5 22.
4
11 Given the functions f : x ↦ 3x 2 1 and g : x ↦ __
x, find the following:
c) (f  g)21
M
y
10
5
pl
12 a) The diagram shows part of the graph
a . The curve
of the function h (x ) 5 _____
x2b
passes through the point A (24, 28).
The vertical line (MN) is an asymptote.
Find the value of: (i) a (ii) b.
d) g  g
pa
g
b) f  g
e
a) f 21
m
�10
5
�5
x
�5
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A
�10
N
b) The graph of h (x ) is transformed as
shown in the diagram right. The point
A is transformed to A9(24, 8). Give
a full geometric description of the
transformation.
M
y
10
A
5
�10
5
�5
x
�5
�10
N
87
2
Functions
13 The graph of y 5 f (x ) is shown in the diagram.
y
2
1
1
�8 �7 �6 �5 �4 �3 �2 �1
2
3
4
5
6
7
8 x
�1
�2
es
a) Make two copies of the coordinate system as shown in the diagram but without the graph
of y 5 f (x ). On the first diagram sketch a graph of y 5 2f (x ), and on the second diagram
sketch a graph of y 5 f (x 2 4).
b) The point A(23, 1) is on the graph of y 5 f (x ). The point A9 is the corresponding point on
the graph of y 5 2f (x ) 2 1. Find the coordinates of A9.
pa
g
14 The diagram below shows the graph of y1 5 f (x). The x-axis is a tangent to f (x ) at
x 5 m and f (x) crosses the x-axis at x 5 n.
e
y
m
n
x
Sa
m
pl
0
y1 � f(x)
On the same diagram, sketch the graph of y2 5 f (x 2 k), where 0 , k , n 2 m and indicate
the coordinates of the points of intersection of y2 with the x-axis.
15 Given functions f : x ↦ x 1 1 and g : x ↦ x3, find the function (f  g)21.
x
16 If f (x) 5 _____ for x  21 and g(x) 5 (f  f )(x), find
x11
a) g(x)
b) (g  g)(2).
________
17 Let f : x ↦
√
1
___
2 2 . Find
2
x
a) the set of real values of x for which f is real and finite;
b) the range of f.
2x 1 1 , x  R, x  1. Find the inverse function, f 21, clearly stating its
18 The function f : x ↦ ______
x21
domain.
88
2x 2 1 .
19 The one-to-one function f is defined on the domain x > 0 by f (x) 5 ______
x12
a) State the range, A, of f.
b) Obtain an expression for f 21(x), for x  A.
20 The function f is defined by f : x ↦ x3.
Find an expression for g(x) in terms of x in each of the following cases
a) (f  g)(x) 5 x 1 1;
b) (g  f )(x) 5 x 1 1.
1
______
21 a) Find the largest set S of values of x such that the function f (x) 5 ________
takes real
√ 3 2 x2
values.
b) Find the range of the function f defined on the domain S.
x11
22 Let f and g be two functions. Given that (f  g)(x) 5 _____ and g(x) 5 2x 2 1,
2
find f (x 2 3).
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23 The diagram below shows the graph of y 5 f (x) which passes through the points A, B,
C and D.
pa
g
Sketch, indicating clearly the images of A, B, C and D, the graphs of
a) y 5 f (x 2 4);
b) y 5 f (2 3x).
y
25
20
pl
15
e
A
10
5
D
B
m
�12�11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0
�5
1
2
3
4
5
6
7
8
9 10 11 12 x
Sa
�10
�15
�20
�25
�30
�35
C
89