Chapter
10
Functions
Syllabus reference: 4.1, 4.2
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Relations and functions
Function notation
Domain and range
Mappings
Linear functions
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A
B
C
D
E
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Contents:
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310
FUNCTIONS
(Chapter 10)
A
RELATIONS AND FUNCTIONS
The charges for parking a car in a short-term car park at an
airport are given in the table alongside.
Car park charges
Period (_t_)
Charge
0 - 1 hours
$5:00
1 - 2 hours
$9:00
2 - 3 hours
$11:00
3 - 6 hours
$13:00
6 - 9 hours
$18:00
9 - 12 hours $22:00
12 - 24 hours $28:00
There is an obvious relationship between the time spent in the
car park and the cost. The cost is dependent on the length of
time the car is parked.
Looking at this table we might ask: How much would be
charged for exactly one hour? Would it be $5 or $9?
To make the situation clear, and to avoid confusion, we could
adjust the table and draw a graph. We need to indicate that
2-3 hours really means a time over 2 hours up to and including
3 hours. So, 2 < t 6 3.
Car park charges
Period
Charge
0 < t 6 1 hours
$5:00
1 < t 6 2 hours
$9:00
2 < t 6 3 hours
$11:00
3 < t 6 6 hours
$13:00
6 < t 6 9 hours
$18:00
9 < t 6 12 hours $22:00
12 < t 6 24 hours $28:00
We now
have:
In mathematical terms, we have a relationship between the two variables time and cost, so
the schedule of charges is an example of a relation.
A relation may consist of a finite number
of ordered pairs, such as f(1, 5), (¡2, 3),
(4, 3), (1, 6)g, or an infinite number of
ordered pairs.
30
The parking charges example is clearly the
latter as any real value of time (t hours)
in the interval 0 < t 6 24 is represented.
10
charge ($)
20
exclusion
inclusion
time (t)
3
9
6
12
15
18
21
24
The set of possible values of the variable on the horizontal axis is called the domain of the
relation.
² ft j 0 < t 6 24g is the domain for the car park relation
For example:
² f¡2, 1, 4g is the domain of f(1, 5), (¡2, 3), (4, 3), (1, 6)g.
The set which describes the possible y-values is called the range of the relation.
² the range of the car park relation is f5, 9, 11, 13, 18, 22, 28g
For example:
² the range of f(1, 5), (¡2, 3), (4, 3), (1, 6)g is f3, 5, 6g.
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We will now look at relations and functions more formally.
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FUNCTIONS
(Chapter 10)
311
RELATIONS
A relation is any set of points on the Cartesian plane.
A relation is often expressed in the form of an equation connecting the variables x and y.
For example, y = x + 3 and x = y2 are the equations of two relations.
These equations generate
sets of ordered pairs.
Their graphs are:
y
y
y=x+3
2
3
x
4
x
-3
x = y2
However, a relation may not be able to be defined by an equation. Below are two examples
which show this:
(1)
y
(2)
All points in the
first quadrant are
a relation.
x > 0, y > 0
x
These 13 points
form a relation.
y
x
FUNCTIONS
A function, sometimes called a mapping, is a relation in which no two
different ordered pairs have the same x-coordinate or first member.
We can see from this definition that a function is a special type of relation.
TESTING FOR FUNCTIONS
Algebraic Tests:
If a relation is given as an equation, and the substitution of any value
for x results in one and only one value of y, we have a function.
For example:
² y = 3x ¡ 1 is a function, as for any value of x there is only one value of y.
² x = y 2 is not a function since if x = 4 then y = §2.
Geometric Test or Vertical Line Test:
If we draw all possible vertical lines on the graph of a relation, the relation:
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² is a function if each line cuts the graph no more than once
² is not a function if at least one line cuts the graph more than once.
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312
FUNCTIONS
(Chapter 10)
Example 1
Self Tutor
Which of the following relations are functions?
a
b
y
c
y
y
x
x
x
a
b
y
c
y
y
x
x
x
a function
a function
not a function
GRAPHICAL NOTE
² If a graph contains a small open circle such as
, this point is not included.
² If a graph contains a small filled-in circle such as
, this point is included.
² If a graph contains an arrow head at an end such as
, then the graph continues
indefinitely in that general direction, or the shape may repeat as it has done previously.
EXERCISE 10A
1 Which of the following sets of ordered pairs are functions? Give reasons.
a f(1, 3), (2, 4), (3, 5), (4, 6)g
b f(1, 3), (3, 2), (1, 7), (¡1, 4)g
c f(2, ¡1), (2, 0), (2, 3), (2, 11)g
d f(7, 6), (5, 6), (3, 6), (¡4, 6)g
e f(0, 0), (1, 0), (3, 0), (5, 0)g
f f(0, 0), (0, ¡2), (0, 2), (0, 4)g
2 Use the vertical line test to determine which of the following relations are functions:
a
b
c
y
d
y
y
x
x
e
y
x
f
g
y
x
h
y
y
y
x
x
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FUNCTIONS
313
(Chapter 10)
3 Will the graph of a straight line always be a function? Give evidence to support your
answer.
4 Give algebraic evidence to show that the relation x2 + y 2 = 9 is not a function.
B
FUNCTION NOTATION
Function machines are sometimes used to illustrate how functions behave.
For example:
x
If 4 is fed into the machine,
2(4) + 3 = 11 comes out.
I double the
input and
then add 3
2x + 3
The above ‘machine’ has been programmed to perform a particular function.
If f is used to represent that particular function we can write:
f is the function that will convert x into 2x + 3.
2 into
2(2) + 3 = 7 and
¡4 into 2(¡4) + 3 = ¡5.
So, f would convert
This function can be written as:
f : x 7! 2x + 3
function f
such that
x is converted into 2x + 3
Two other equivalent forms we use are: f (x) = 2x + 3 or y = 2x + 3
f (x) is the value of y for a given value of x, so y = f (x).
Notice that for f(x) = 2x + 3, f(2) = 2(2) + 3 = 7 and f(¡4) = 2(¡4) + 3 = ¡5:
Consequently,
f (2) = 7 indicates that the point (2, 7)
lies on the graph of the function.
f (¡4) = ¡5 indicates that the point
(¡4, ¡5) also lies on the graph.
Likewise,
y
(2, 7)
f(x) = 2x + 3
3
x
3
Note:
² f (x) is read as “f of x”.
(-4,-5)
² f is the function which converts x into f (x),
so we write f : x 7! f (x):
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² y = f (x) is sometimes called the image of x.
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314
FUNCTIONS
(Chapter 10)
Example 2
Self Tutor
If f : x 7! 2x2 ¡ 3x, find the value of:
a f (5)
b f(¡4)
f (x) = 2x2 ¡ 3x
a f(5) = 2(5)2 ¡ 3(5)
= 2 £ 25 ¡ 15
= 35
freplacing x with (5)g
b f(¡4) = 2(¡4)2 ¡ 3(¡4)
= 2(16) + 12
= 44
freplacing x with (¡4)g
Example 3
Self Tutor
If f (x) = 5 ¡ x ¡ x2 , find in simplest form:
a f (¡x)
b f (x + 2)
a f(¡x) = 5 ¡ (¡x) ¡ (¡x)2
= 5 + x ¡ x2
freplacing x with (¡x)g
b f(x + 2) = 5 ¡ (x + 2) ¡ (x + 2)2
= 5 ¡ x ¡ 2 ¡ [x2 + 4x + 4]
= 3 ¡ x ¡ x2 ¡ 4x ¡ 4
= ¡x2 ¡ 5x ¡ 1
freplacing x with (x + 2)g
EXERCISE 10B
1 If f : x 7! 3x + 2, find the value of:
d f(¡5)
e f (¡ 13 )
c f (¡3)
d f(¡7)
e f ( 32 )
4
, find the value of:
x
b g(4)
c g(¡1)
d g(¡4)
¡ ¢
e g ¡ 12
a f (0)
b f (2)
c f (¡1)
2 If f : x 7! 3x ¡ x2 + 2, find the value of:
a f (0)
b f (3)
3 If g : x 7! x ¡
a g(1)
4 If f (x) = 7 ¡ 3x, find in simplest form:
a f (a)
c f (a + 3)
b f(¡a)
d f (b ¡ 1)
e f (x + 2)
f f (x + h)
5 If F (x) = 2x2 + 3x ¡ 1, find in simplest form:
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d F (x2 )
c F (¡x)
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a F (x + 4) b F (2 ¡ x)
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FUNCTIONS
6 Suppose G(x) =
a Evaluate:
2x + 3
:
x¡4
i G(2)
ii G(0)
(Chapter 10)
315
¡ ¢
iii G ¡ 12
b Find a value of x such that G(x) does not exist.
c Find G(x + 2) in simplest form.
d Find x if G(x) = ¡3:
7 f represents a function. What is the difference in meaning between f and f (x)?
8 The value of a photocopier t years after purchase is
given by V (t) = 9650 ¡ 860t euros.
a Find V (4) and state what V (4) means.
b Find t when V (t) = 5780 and explain what this
represents.
c Find the original purchase price of the photocopier.
9 On the same set of axes draw the graphs of three different functions f (x) such that
f (2) = 1 and f (5) = 3.
10 Find a linear function f(x) = ax + b for which f (2) = 1 and f (¡3) = 11:
11 Find constants a and b if f (x) = ax +
b
, f (1) = 1, and f (2) = 5:
x
12 Given T (x) = ax2 +bx+c, find a, b and c if T (0) = ¡4, T (1) = ¡2 and T (2) = 6:
13 If f(x) = 2x , show that f (a)f (b) = f (a + b):
C
DOMAIN AND RANGE
The domain of a relation is the set of permissible values that x may have.
The range of a relation is the set of permissible values that y may have.
The domain and range of a relation are often described using interval notation.
For example:
y
All values of x > ¡1 are permissible.
So, the domain is fx j x > ¡1g.
All values of y > ¡3 are permissible.
So, the range is fy j y > ¡3g.
(1)
x
(-1,-3)
y
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x can take any value.
So, the domain is fx j x 2 R g.
y cannot be > 1.
So, the range is fy j y 6 1g.
(2, 1)
100
(2)
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316
FUNCTIONS
(Chapter 10)
x can take all values except x = 2.
So, the domain is fx j x 6= 2g.
Likewise, the range is fy j y 6= 1g.
y
(3)
y=1
x
profit ($)
x=2
100
(4) For the profit function alongside:
range
² the domain is fx j x > 0g
items made (x)
² the range is fy j y 6 100g.
10
domain
Click on the icon to obtain software for finding the domain and range of different
functions.
Example 4
DOMAIN
AND RANGE
Self Tutor
For each of the following graphs, state the domain and range:
a
b
y
y
(4, 3)
x
x
(8, -2)
(2, -1)
fx j x 6 8g
fy j y > ¡2g
a Domain is
Range is
b Domain is
Range is
fx j x 2 R g
fy j y > ¡1g
EXERCISE 10C
1 For each of the following graphs, find the domain and range:
a
(-1, 3)
b
y
c
y
y
(5, 3)
(-1, 1)
x
y = -1
x
x
x=2
d
e
y
f
y
y
(Qw , 6_Qr )
(0, 2)
x
x
x
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(1, -1)
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FUNCTIONS (Chapter 10)
g
h
y
i
y
317
y
(-1, 2)
x
(2, -2)
y = -2
(-4, -3)
x
-1
x
x = -2
x=2
2 For each of the following functions, state the values of x for which f (x) is undefined:
p
1
¡7
b f : x 7! 2
c f (x) = p
a f (x) = x + 6
x
3 ¡ 2x
3 Use technology to help sketch graphs of the following functions. Find the
domain and range of each.
p
1
b f : x 7! 2
a f (x) = x
x
p
c f : x 7! 4 ¡ x
d y = x2 ¡ 7x + 10
1
e f : x 7! 5x ¡ 3x2
f f : x 7! x +
x
x+4
g y=
h y = x3 ¡ 3x2 ¡ 9x + 10
x¡2
3x ¡ 9
j y = x2 + x¡2
i f : x 7! 2
x ¡x¡2
1
k y = x3 + 3
l f : x 7! x4 + 4x3 ¡ 16x + 3
x
D
DOMAIN
AND RANGE
MAPPINGS
In the previous section, we introduced functions as
‘machines’ which convert x into f (x).
pp
a
For example, the function f : x 7! 5x + 2 maps x
onto ‘two more than five lots of x’.
f(x)
xm
The function f : x 7! f (x) maps elements from the
domain of the function onto elements in the range
of the function.
domain
(input)
ed
onto
f(x)
range
(output)
We can specify a domain to limit the input possibilities.
For the domain fx j 1 6 x 6 4, x 2 Z g, the permissible values of f (x) are restricted to:
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f (1) = 5(1) + 2 = 7
f (2) = 5(2) + 2 = 12
f (3) = 5(3) + 2 = 17
f (4) = 5(4) + 2 = 22
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FUNCTIONS (Chapter 10)
f : x 7! 5x + 2
We can represent f : x 7! 5x + 2 for this
domain using a mapping diagram:
1
2
3
4
7
12
17
22
domain
range
In this case each element in the domain f1, 2, 3, 4g corresponds to a single element in the
range f7, 12, 17, 22g.
For a relation:
² multiple elements in the domain may correspond to the same element in the range.
² an element in the domain may have more than one corresponding element in the range.
For a function, however, each element in the domain corresponds to exactly one element in
the range.
Example 5
Self Tutor
Construct a mapping diagram for f : x 7! x2 on the domain f¡2, ¡1, 0, 1, 2g.
Is this relation a function? Display the results on a set of axes.
x
¡2
¡1
0
1
2
f (x)
4
1
0
1
4
-2
-1
0
1
2
4
1
0
This relation is a function since each element in
the domain corresponds to exactly one element in
the range.
y
2
-1
1
x
Example 6
Self Tutor
p
Construct a mapping diagram for p : x 7! § x on the domain f0, 1, 4, 9g.
Is this relation a function? Display the results on a set of axes.
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IB_STSL-2ed
FUNCTIONS
319
(Chapter 10)
y
The mapping diagram shows
single elements of the domain
mapping to multiple elements
within the range. This relation is
not a function.
2
5
x
-2
EXERCISE 10D
1 For the following relations on ¡2 6 x 6 2, x 2 Z :
i
ii
iii
iv
draw a mapping diagram to represent the relation
list the elements of the range using set notation
determine if the relation is a function
illustrate the relation on a set of axes.
b f(x) = x2 + 1
a f (x) = 3x ¡ 1
c f (x) = 3 ¡ 4x
2
d f : x 7! 3 § x
f f (x) = x3
x+3
i f (x) =
, x 6= 0
x
e f(x) = 2x ¡ x + 1
1
h f : x 7!
x+3
g f (x) = 3x
Example 7
Self Tutor
The diagram shows a function f mapping
members of set X to members of set Y .
a Using set notation, write down the members
of the domain and range.
b Find the equation of the function f .
f :X!Y
-2
0
1
2
-1
-7
-1
2
5
-4
a Domain = f¡2, ¡1, 0, 1, 2g, Range = f¡7, ¡4, ¡1, 2, 5g
b We use the mapping diagram to construct
a table of values:
x
¡2
¡1
0
1
2
f (x)
¡7
¡4
¡1
2
5
y
4
2
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Plotting the points on a set of axes, we see
that the points form a straight line.
So, f (x) = mx + c where
5¡2
fusing (2, 5) and (1, 2)g
m=
2¡1
) m=3
The y-intercept c = ¡1.
) f (x) = 3x ¡ 1 or f : x 7! 3x ¡ 1
on the domain fx j ¡2 6 x 6 2, x 2 Z g.
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IB_STSL-2ed
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FUNCTIONS
(Chapter 10)
2 In the following mapping diagrams, the function f maps the elements of X onto the
elements of Y .
i Use set notation to write down the domain of f .
ii Use set notation to write down the range of f .
iii Find the equation of the function f.
a
X
b
-1
0
2
-2
1
3
5
9
1
7
6
-3
9
0
3
13
-23
25
-11
1
Y
X
c
0
2
1
4
3
5
3
4
1
2
Y
d
X
Y
X
-3
3
15
9
6
13
1
-23
-11
-5
Y
3 In the following mapping diagrams, the function f maps X onto Y .
i Sketch the function f on a set of axes.
ii Find the equation of the function f.
a
b
16
1
4
3
2
1
0
3
1
2
4
1
8
2
4
16
9
4
X
Y
X
c
-2
2
-1
1
0
4
1
0
-2
2
-1
1
0
7
4
3
Y
d
X
Y
X
Y
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To find the equations of the
functions, it may help to compare the
graphs with those in question 1.
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FUNCTIONS
E
(Chapter 10)
321
LINEAR FUNCTIONS
A linear function is a function of the form f (x) = ax + b or f : x 7! ax + b
where a and b are constants, a 6= 0.
GRAPHS OF LINEAR FUNCTIONS
y
Consider the function f (x) = ax + b.
The graph of y = f (x) is a straight line with
gradient a and y-intercept b.
a
x-intercept
b
The x-intercept is where the graph cuts the x-axis.
It is found by solving the equation f (x) = 0.
1
x
INVESTIGATION
BAMBOO
Bamboo is the fastest growing plant in the world, with some species growing
up to 1 metre per day.
Xuanyu planted a 30 cm high bamboo plant in her garden bed. She found
that with consistent weather it grew 10 cm each day.
What to do:
1 Copy and complete this table of values which gives the height H of the bamboo
after t days.
t (days)
0
1
H (cm)
30
40
2
3
4
2 Plot the points on a set of axes and connect them
with straight line segments.
5
6
H (cm)
3 Explain why it is reasonable to connect the points
with straight line segments.
4 Discuss whether it is reasonable to continue the line
for t < 0 and for t > 6 days. Hence state the
domain of the function H(t):
5 Find the ‘H-intercept’ and gradient of the line.
t (days)
6 Find an equation for the function H(t).
7 Use your equation to find the value of H(10). Explain what this value represents.
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8 How long will it take for the bamboo to be 1 m high?
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FUNCTIONS
(Chapter 10)
Example 8
Self Tutor
The cost of hiring a tennis court is given by the formula C(h) = 5h + 8 dollars where
h is the number of hours the court is hired for. Find the cost of hiring the tennis court
for:
a 4 hours
b 10 hours.
The formula is C(h) = 5h + 8.
a Substituting h = 4 we get
C(4) = 5(4) + 8
= 20 + 8
= 28
It costs $28 to hire the court for
4 hours.
b Substituting h = 10 we get
C(10) = 5(10) + 8
= 50 + 8
= 58
It costs $58 to hire the court for
10 hours.
You can use a graphics calculator or graphing package to help you in the
following exercise.
GRAPHING
PACKAGE
EXERCISE 10E
1 The cost of staying at a hotel is given by the formula C(d) = 50d + 20 euros where d
is the number of days a person stays. Find the cost of staying for:
a 3 days
b 6 days
c 2 weeks
2 The thermometer on Charlotte’s kitchen oven uses the Celsius scale, but her recipe book
gives the required temperature on the Fahrenheit scale. The formula which links the two
temperature scales is TC (F ) = 59 (F ¡ 32) where TC is the temperature in degrees
Celsius and F is the temperature in degrees Fahrenheit.
Convert the following Fahrenheit temperatures into Celsius:
a 212o F
b 32o F
c 104o F
d 374o F
3 The value of a car t years after its purchase is given by V (t) = 25 000 ¡ 3000t pounds.
a Find V (0) and state the meaning of V (0):
b Find V (3) and state the meaning of V (3):
c Find t when V (t) = 10 000 and explain what this represents.
4 Find the linear function f (x) = ax + b such that f (2) = 7 and f (¡1) = ¡5.
Example 9
Self Tutor
Ace taxi services charge $3:30 for stopping to pick up a passenger and then $1:75 for
each kilometre of the journey.
a Copy and complete:
Distance (d km)
0
2
4
6
8
10
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FUNCTIONS
(Chapter 10)
323
b Graph C against d.
c Find the function C(d) which connects the variables.
d Find the cost of a 9:4 km journey.
a
Distance (d km)
0
2
4
6
8
10
Cost ($C)
3:30
6:80
10:30
13:80
17:30
20:80
adding 2 £ $1:75 = $3:50 each time.
b
c The ‘y-intercept’ is 3:30 and
20:80 ¡ 17:30
the gradient =
10 ¡ 8
= 1:75
) C(d) = 1:75d + 3:3
C ($)
20
15
10
5
d (km)
2
4
6
8
10
d C(9:4) = 1:75 £ 9:4 + 3:3
= 19:75
12
) the cost is $19:75
5 An electrician charges $60 for calling and $45 per hour he spends on the job.
a From a table of values, plot the amount $C the electrician charges against the
hours t he works for t = 0, 1, 2, 3, 4 and 5.
b Use your graph to determine the cost function C in terms of t.
c Use the cost function to determine the electrician’s total cost for a job lasting 6 12
hours. Use your graph to check your answer.
6 A rainwater tank contains 265 litres. The tap is left on and 11 litres escape per minute.
a Construct a table of values for the volume V litres left in the tank after t minutes
for t = 0, 1, 2, 3, 4 and 5.
b Use your table to graph V against t.
c Use your graph to determine the function V (t).
d Use your function to determine:
i how much water is left in the tank after
15 minutes
ii the time taken for the tank to empty.
e Use your graph to check your answers to d i and d ii.
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7 The cost of running a truck is E158 plus E365 for every one thousand kilometres driven.
a Without graphing, determine the cost EC in terms of the number of thousands of
kilometres n.
b Find the cost of running the truck a distance of 3750 km.
c How far could the truck travel if E5000 was available?
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FUNCTIONS
(Chapter 10)
8 A salesperson’s wage is determined from the
graph alongside.
weekly wage ($)
400
a Determine the weekly wage $W in terms
of the sales $s thousand dollars.
b Find the weekly wage if the salesperson
made $33 500 worth of sales.
c Determine the sales necessary for a
weekly wage of $830.
300
200
100
sales (1000s of dollars)
3
2
4
1
Example 10
Self Tutor
An appliance manufacturer can set up the machinery to produce a new line of toaster
for $12 000. Following this initial setup cost, every 100 toasters produced will cost a
further $1000. The toasters are then sold to a distributor for $25 each.
a Determine the cost of production function C(n) where n is the number of toasters
manufactured.
b Determine the income or revenue function R(n):
c Graph C(n) and R(n) on the same set of axes for 0 6 n 6 1500.
d How many toasters need to be produced and sold in order to ‘break even’?
e Calculate the profit or loss made when
i 400 toasters
ii 1500 toasters are produced and sold.
a After the fixed cost of $12 000, each toaster costs $10 to produce.
) C(n) = 10n + 12 000 dollars
fgradient = cost/item = $10g
b Each toaster is sold for $25, so R(n) = 25n dollars.
c
40
$ (’000)
R(n)
30
profit
(15, 37.5)
(15, 27)
C(n)
20
loss
10
n (’00 toasters)
5
loss made
10
profit made
15
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d To the left of the point of intersection, C(n) > R(n), so a loss is made.
The manufacturer ‘breaks even’ where C(n) = R(n)
) 10n + 12 000 = 25n
) 12 000 = 15n
) n = 800
800 toasters must be produced and sold in order to ‘break even’.
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FUNCTIONS
e
Profit = R(n) ¡ C(n)
) P (n) = 25n ¡ (10n + 12 000)
= 15n ¡ 12 000
(Chapter 10)
325
i P (400) = 15 £ 400 ¡ 12 000
= ¡$6000
This is a loss of $6000.
ii P (1500) = 15 £ 1500 ¡ 12 000
= $10 500
This is a profit of $10 500.
9 Self adhesive label packs are produced with cost function C(n) = 3n + 20 dollars and
revenue function R(n) = 5n + 10 dollars where n is the number of packs produced.
a Graph each function on the same set of axes, clearly labelling each graph.
b Determine the number of packs which must be produced and sold to ‘break even’.
Check your answer algebraically.
c For what values of n is a profit made?
d How many self adhesive label packs need to be produced and sold to make a profit
of $100?
10 Two way adaptors sell for $7 each. The adaptors cost $2:50 each to make with fixed
costs of $300 per day regardless of the number made.
a Find revenue and cost functions in terms of the number n of adaptors manufactured
per day.
b Graph the revenue and cost functions on the same set of axes.
c Determine the ‘break even’ level of production and check your answer algebraically.
d How many adaptors must be made and sold every day to make a profit of $1000?
11 A new novel is being printed. The plates
required for the printing process cost E6000,
after which the printing costs E3250 per
thousand books. The books are to sell at
E9:50 each with an unlimited market.
a Determine cost and revenue functions for
the production of the novel.
b Graph the cost and revenue functions on the same set of axes.
c How many books must be sold in order to ‘break even’? Check your answer
algebraically.
d What level of production and sale will produce a E10 000 profit?
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12 Waverley Manufacturing produces carburettors for motor vehicles. Each week there is
a fixed cost of $2100 to keep the factory running. Each carburettor costs $13:20 in
materials and $14:80 in labour to produce. Waverley is able to sell the carburettors to
the motor vehicle manufacturers at $70 each.
a Determine Waverley’s cost and revenue functions in terms of the number n
manufactured per week.
b Draw graphs of the cost and revenue functions on the same set of axes. Use your
graph to find the ‘break even’ point.
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FUNCTIONS
(Chapter 10)
c Find the expression for the profit function and check your answer for the ‘break
even’ value of n.
d Use your profit function to find:
i the weekly profit for producing and selling 125 carburettors
ii the number of carburettors required to make a profit of at least $1300.
REVIEW SET 10A
1 If f (x) = 2x ¡ x2 find:
a f (2)
c f(¡ 12 )
b f(¡3)
2 If f (x) = ax + b where a and b are constants, find a and b for f(1) = 7 and
f (3) = ¡5.
3 If g(x) = x2 ¡ 3x, find in simplest form:
b g(x2 ¡ 2)
a g(x + 1)
4 For each of the following graphs determine:
i the range and domain
ii whether it is a function.
a
b
y
y
1
-1
5
-Wl_T
x
-1
x
-3
(2,-5)
2
.
x2
a For what value of x is f (x) meaningless?
5 Consider f (x) =
b Sketch the graph of this function using technology.
c State the domain and range of the function.
a Construct a mapping diagram for f : x 7! x2 ¡ 4 on the domain
f¡2, ¡1, 0, 1, 2g.
6
b List the elements of the range of f using set notation.
c Determine whether f is a function.
d Draw the graph of f.
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7 A marquee hire company charges $130 for setting up and packing down, and $80
per day of use.
a Construct a table of values showing the cost $C of hiring a marquee for d days,
where d = 0, 1, 2, 3, 4.
b Use your table to graph C against d.
c Use your graph to find the function linking C and d.
d The company offers a special deal in which the total cost of a week’s hire is
$650. How much money does this save?
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327
(Chapter 10)
REVIEW SET 10B
1 Use the vertical line test to determine which of the following relations are functions:
a
b
c
y
y
y
x
x
x
2 If f : x 7! 3x ¡ 5, find:
a f (2)
b f (0)
d f (x2 ¡ x)
c f (x + 1)
3 Find constants a and b if f (x) =
a
+ b, f (1) = ¡4 and f (5) = 0.
x¡2
4 For each of the following graphs, find the domain and range:
a
b
y
y
y = (x-1)(x-5)
(1,-1)
x
x
x=2
(3, -4)
5 For each of the following functions, state the values of x for which f(x) is
undefined:
p
1
a f (x) = 3
b f : x 7! x ¡ 2
c f (x) = 3x + 7
x
6 For the following functions on ¡2 6 x 6 2 where x 2 Z :
i draw a mapping diagram to represent f (x)
ii list the elements of the domain of f (x) using set notation
iii list the elements of the range of f (x) using set notation.
b f (x) = x2 ¡ x + 2
a f : x 7! 2x + 5
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7 The amount of oil O left in a leaky barrel
after t minutes is shown on the graph
alongside.
a Express the amount of oil left after
t minutes as a formula.
b How much oil is left after 15 minutes?
c How long will it take before:
i there are only 50 litres of oil left
ii the barrel is empty?
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FUNCTIONS (Chapter 10)
REVIEW SET 10C
1 Which of these sets of ordered pairs are functions? Give reasons for your answers.
a f(1, 2), (¡1, 2), (0, 5), (2, ¡7)g
b f(0, 1), (1, 3), (2, 5), (0, 7)g
c f(6, 1), (6, 2), (6, 3), (6, 4)g
2 For each of the following graphs, find the domain and range:
a
b
y
y
(3, 2)
3
(3, 1)
1
(-2, 1)
x
x
-1
3 Draw a possible graph of a function g(x) where g(2) = 5, g(4) = 7 and g(6) = ¡1.
4 Find a, b and c if f (0) = 5, f (¡2) = 21, f (3) = ¡4 and f (x) = ax2 + bx + c.
5 For each of the following graphs determine:
i the domain and range
a
ii whether it is a function.
b
y
y
2
3
x
x
3
-1
-2
y = -1
x=1
2
-2
6 In the following mapping diagrams, the function f maps X onto Y .
i Use set notation to write down the domain and range of f .
ii Sketch the function f on a set of axes.
iii Find the equation of the function f .
a
b
X
5
-1
2
8
11
2
0
1
3
4
Y
X
-2
-1
0
1
2
3
Y
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7 Tennis balls are sold in packs of 3 for $11 per pack. The balls cost $2:40 each to
produce with factory running costs of $1750 per day.
a Find the cost and revenue functions for making n tennis balls in a day.
b Graph the cost and revenue functions on the same set of axes.
c Determine the ‘break even’ production level. Check your answer algebraically.
d How many tennis ball packs must be sold to make $400 profit per day?
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