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Functions
Still glides the stream and shall forever
glide; The form remains, the function
never dies. William Wordsworth
Why fly to Geneva in January?
Several people arriving at Geneva airport from London were asked the
main purpose of their visit. Their answers were recorded.
David
Joanne Skiing
Jonathan
Returning home
Louise
To study abroad
Paul
This is an example of a mapping.
Shamaila
Karen
Business
106
The language of functions
A mapping is any rule which associates two sets of items. In this example, each
of the names on the left is an object , or input , and each of the reasons on the right
is an image , or output.
For a mapping to make sense or to have any practical application, the inputs
and outputs must each form a natural collection or set. The set of possible
inputs (in this case, all of the people who flew to Geneva from London in
January) is called the domain of the mapping.
The seven people questioned in this example gave a set of four reasons, or
outputs. These form the range of the mapping for this particular set of
inputs.
Notice that Jonathan, Louise and Karen are all visiting Geneva on business: each
person gave only one reason for the trip, but the same reason was given by several
people. This mapping is said to be many-to-one. A mapping can also be one-to-
one , one-to-many or many-to-many. The relationship between the people from
any country and their passport numbers will be one-to-one. The relationship
between the people and their items of luggage is likely to be one-to-many, and
that between the people and the countries they have visited in the last 10 years
will be many-to-many.
Mappings
In mathematics, many (but not all) mappings can be expressed using algebra.
Here are some examples of mathematical mappings.
4
(a) Domain: integers Range
Objects Images
General rule: x 2 x 5
(b) Domain: integers Range
Objects Images
π
General rule: Rounded whole numbers Unrounded numbers
(c) Domain: real numbers Range
Objects Images
General rule: x ° sin x °
(d) Domain: quadratic Range
equations with real roots
Objects Images
x
2
4 x 3 0 0
x
2
x 0 1
x
2
3 x 2 0 2
General rule: ax
2
bx c 0
x
x
b
2
4 ac 2 a
b
2
4 ac 2 a
The
language
of
functions
P
1
This means x is a positive real number.
This means x is a natural number, i.e. a positive integer or zero.
y = 3x + 2, x
y = 3x + 2, x
The open circle shows that (0, 2) is not part of the line.
y
4
3
2
1
Sketch the graph of y 3 x 2 when the domain of x is
(i) x
4
(ii) x
(iii) x .
SOLUTION
(i) When the domain is , all values of y are possible. The range is therefore , also.
(ii) When x is restricted to positive values, all the values of y are greater than 2,
so the range is y 2.
(iii) In this case the range is the set of points {2, 5, 8, …}. These are clearly all
of the form 3 x 2 where x is a natural number (0, 1, 2, …). This set can be
written neatly as {3 x 2 : x }.
y y y
y = 3 x + 2, x
O x O x O x
Figure 4.
When you draw the graph of a mapping, the x co-ordinate of each point is an
input value, the y co-ordinate is the corresponding output value. The table
below shows this for the mapping x x
2
, or y x
2
, and figure 4.2 shows the
resulting points on a graph.
–2 –1 0 1 2
x
Figure 4.
If the mapping is a function, there is one and only one value of y for every value
of x in the domain. Consequently the graph of a function is a simple curve or
line going from left to right, with no doubling back.
EXAMPLE 4.
The
language
of
functions
P
1
Input
( x )
Output
( y )
Point
plotted
2 4
(2, 4)
1 1
(1, 1)
0 0 (0, 0)
1 1 (1, 1)
2 4 (2, 4)
4 P y y = 2x + 1 O x y y = x3 – x
O 1 x y 1 y = ±2x O x
y 5 y = ± 25 – x
O 5 x domain: –5 x 5
Figure 4.3 illustrates some different types of mapping. The graphs in (a) and (b) illustrate functions, those in (c) and (d) do not. (a) One-to-one (b) Many-to-one (c) One-to-many (d) Many-to-many Figure 4. 1 Describe each of the following mappings as either one-to-one, many-to- one, one-to-many or many-to-many, and say whether it represents a function. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) EXERCISE 4A Functions
P
Composite functions
It is possible to combine functions in several different ways, and you have
already
4
met some of these. For example, if f( x ) x
2
and g( x ) 2 x , then you could write
f( x ) g( x ) x
2
2 x.
In this example, two functions are added.
Similarly if f( x ) x and g( x ) sin x , then f( x ).g( x )
x sin x.
In this example, two functions are multiplied.
Sometimes you need to apply one function and then apply another to the answer.
You are then creating a composite function or a function of a function.
A new mother is bathing her baby for the first time. She takes the temperature
of the bath water with a thermometer which reads in Celsius, but then has to
convert the temperature to degrees Fahrenheit to apply the rule that her own
mother taught her:
At one o five
He’ll cook alive
But ninety four
is rather raw.
Write down the two functions that are involved, and apply them to readings of
(i) 30°C (ii) 38°C (iii) 45°C.
SOLUTION
The first function converts the Celsius temperature C into a
Fahrenheit temperature, F.
F
C
The second function maps Fahrenheit temperatures on to the state of the bath.
F 94 too cold
94 F 105 all right
F 105 too hot
This gives
(i) 30°C 86°F too cold
(ii) 38°C 100.4°F all right
(iii) 45°C 113°C too hot.
EXAMPLE 4.
Function s
g f
Read this as ‘g of f of x’.
(or gf(x)).
f
domain of g
g
domain of f range of f range of gf
In this case the composite function would be (to the nearest degree)
P
1
C 34°C too cold
35°C C 40°C all right
4
C 41°C too hot.
In algebraic terms, a composite function is constructed as
f
Input x Output f( x )
g
Input f( x ) Output g[f( x )]
Thus the composite function gf( x ) should be performed from right to left:
start with x then apply f and then g.
Notation
To indicate that f is being applied twice in succession, you could write ff( x ) but
you would usually use f
2
( x ) instead. Similarly g
3
( x ) means three applications of g.
In order to apply a function repeatedly its range must be completely contained
within its domain.
Order of functions
If f is the rule ‘square the input value’ and g is the rule ‘add 1’, then
f g
x x
2
x
2
square add 1
So gf( x ) x
2
Notice that gf( x ) is not the same as fg( x ), since for fg( x ) you must apply g first. In
the example above, this would give:
x ( x 1) ( x 1)
2
add 1 square
and so fg( x ) ( x 1)
2
Clearly this is not the same result.
Figure 4.4 illustrates the relationship between the domains and ranges of the
functions f and g, and the range of the composite function gf.
gf
Figure 4.
113
functions Composite
This is a short way of writing x is an integer.
Inverse functions
P
Look at the mapping x x 2 with domain the set of integers.
Domain Range
4
x x 2
The mapping is clearly a function, since for every input there is one and only
one output, the number that is two greater than that input.
This mapping can also be seen in reverse. In that case, each number maps on to
the number two less than itself: x x 2. The reverse mapping is also a function
because for any input there is one and only one output. The reverse mapping is
called the inverse function , f
1
Function: f : x x 2 x .
Inverse function: f
1
: x x 2 x .
For a mapping to be a function which also has an inverse function, every
object in the domain must have one and only one image in the range, and
vice versa. This can only be the case if the mapping is one-to-one.
So the condition for a function f to have an inverse function is that, over the
given domain, f represents a one-to-one mapping. This is a common situation,
and many inverse functions are self-evident as in the following examples, for all
of which the domain is the real numbers.
f : x x 1; f
1
: x x 1
g : x 2 x ; g
1
: x
x
h : x x
3
; h
1
: x
3
x
●
? Some of the following mappings are functions which have inverse functions, and
others are not.
(a) Decide which mappings fall into each category, and for those which do
not have inverse functions, explain why.
(b) For those which have inverse functions, how can the functions and
their inverses be written down algebraically?
functions Inverse
4 P y y = f(x) O
1 x y y = g(x) O x y y = h(x) O x f(x) f(x) = x 4
x (i) Temperature measured in Celsius temperature measured in Fahrenheit. (ii) Marks in an examination grade awarded. (iii) Distance measured in light years distance measured in metres. (iv) Number of stops travelled on the London Underground fare. You can decide whether an algebraic mapping is a function, and whether it has an inverse function, by looking at its graph. The curve or line representing a one- to-one function does not double back on itself and has no turning points. The x values cover the full domain and the y values give the range. Figure 4. illustrates the functions f, g and h given on the previous page. Figure 4. Now look at f( x ) x 2 for x (figure 4.6). You can see that there are two distinct input values giving the same output: for example f(2) f(2) 4. When you want to reverse the effect of the function, you have a mapping which for a single input of 4 gives two outputs, 2 and 2. Such a mapping is not a function. Figure 4. You can make a new function, g( x ) x 2 by restricting the domain to (the set of positive real numbers). This is shown in figure 4.7. The function g( x ) is a one-to-one function and its inverse is given by g 1 ( x ) means ‘the positive square root of’. since the sign Functions
, x y A(0, 4)y = x C(–4, 2) B(–1, 1) A'(4, 0) x B'(1, –1) C'(2, –4) 4 P domain of f and range of f–1 range of f and domain of f– This result can be used to obtain a sketch of the inverse function without having to find its equation, provided that the sketch of the original function uses the same scale on both axes. Figure 4. Finding the algebraic form of the inverse function To find the algebraic form of the inverse of a function f( x ), you should start by changing notation and writing it in the form y …. Since the graph of the inverse function is the reflection of the graph of the original function in the line y x , it follows that you may find its equation by interchanging y and x in the equation of the original function. You will then need to make y the subject of your new equation. This procedure is illustrated in Example 4.4. Find f 1 ( x ) when f( x ) 2 x 1, x . SOLUTION The function f( x ) is given by y 2 x 1 Interchanging x and y gives x 2 y 1 Rearranging to make y the subject: y x – 1 So f 1 ( x ) x
Sometimes the domain of the function f will not include the whole of . When any real numbers are excluded from the domain of f, it follows that they will be excluded from the range of f 1 , and vice versa. f 118 Figure 4. f
EXAMPLE 4. Function s
x 2
Find f
1
( x ) when f( x ) 2 x 3 and the domain of f is x 4.
P
4
Rearranging the inverse function to make y the subject: y
x
The full definition of the inverse function is therefore:
f
1
( x )
x
for x 5.
y
y = f( x )
y = x
(4, 5)
(5, 4)
y = f
( x )
O
x
Figure 4.
You can see in figure 4.10 that the inverse function is the reflection of a
restricted part of the line y 2 x 3.
(i) Find f
1
( x ) when f( x ) x
2
2, x 0.
(ii) Find f(7) and f
1
f(7). What do you notice?
SOLUTION
(i) Domain Range
Function: y x
2
2 x 0 y 2
Inverse function: x y
2
2 x 2 y 0
Rearranging the inverse function to make y its subject: y
2
x 2.
This gives y ± x 2, but since you know the range of the inverse function
to be y 0 you can write:
y or just y x 2.
EXAMPLE 4.
EXAMPLE 4.
functions Inverse
SOLUTION
Doma
in
Range
Function: y 2 x 3 x 4 y 5
Inverse function: x 2 y 3 x 5 y 4
6 (i) Show that x
2
4 x 7 ( x 2)
2
a , where a is to be determined.
(ii) Sketch the graph of y x
2
4 x 7, giving the equation of its axis of
symmetry and the co-ordinates of its vertex.
4
The function f is defined by f: x x
2
4 x 7 with domain the set of all real
numbers.
(iii) Find the range of f.
(iv) Explain, with reference to your sketch, why f has no inverse with its
given domain. Suggest a domain for f for which it has an inverse.
7 The function f is defined by f : x 4 x
3
3, x .
Give the corresponding definition of f
1
State the relationship between the graphs of f and f
1
[MEI]
[UCLES]
8 Two functions are defined for x as f( x ) x
2
and g( x ) x
2
4 x 1.
(i) Find a and b so that g( x ) f( x a ) b.
(ii) Show how the graph of y g( x ) is related to the graph of y f( x ) and
sketch the graph of y g( x ).
(iii) State the range of the function g( x ).
(iv) State the least value of c so that g( x ) is one-to-one for x c.
(v) With this restriction, sketch g( x ) and g
1
( x ) on the same axes.
9 The functions f and g are defined for x by f :
x 4 x 2 x
2
g : x 5 x 3.
(i) Find the range of f.
(ii) Find the value of the constant k for which the equation gf( x ) = k
has equal roots.
[Cambridge AS & A Level Mathematics 9709, Paper 12 Q3 June 2010]
10 Functions f and g are defined by
f : x k – x for x , where k is a constant,
g : x
x 2
for x , x –2.
(i) Find the values of k for which the equation f( x ) = g( x ) has two
equal roots and solve the equation f( x ) = g( x ) in these cases.
(ii) Solve the equation fg( x ) = 5 when k = 6.
(iii) Express g
( x ) in terms of x.
[Cambridge AS & A Level Mathematics 9709, Paper 1 Q11 June 2006]
4B Exercise
P
1
11 The function f is defined by f : x 2 x
2
(i) Express f( x ) in the form a ( x + b )
2
- c , where a , b and c are
constants.
4
(ii) State the range of f.
(iii) Explain why f does not have an inverse.
The function g is defined by g : x 2 x
2
- 8 x + 11 for x A , where A is a
constant.
(iv) State the largest value of A for which g has an inverse.
(v) When A has this value, obtain an expression, in terms of x , for
g
( x ) and state the range of g
[Cambridge AS & A Level Mathematics 9709, Paper 1 Q
November 2007]
12 The function f is defined by f : x 3 x – 2 for x .
(i) Sketch, in a single diagram, the graphs of y = f( x ) and y = f
1
( x ), making clear the relationship between the two graphs.
The function g is defined by g : x 6 x – x
2
for x .
(ii) Express gf( x ) in terms of x , and hence show that the
maximum value of gf( x ) is 9.
The function h is defined by h : x 6 x – x
2
for x 3.
(iii) Express 6 x – x
2
in the form a – ( x – b )
2
, where a and b
are positive constants.
(iv) Express h
( x ) in terms of x.
[Cambridge AS & A Level Mathematics 9709, Paper 1 Q
November 2008]
KEY POINTS
1 A mapping is any rule connecting input values (objects) and output
values (images). It can be many-to-one, one-to-many, one-to-one
or many-to-many.
2 A many-to-one or one-to-one mapping is called a function. It is a mapping
for which each input value gives exactly one output value.
3 The domain of a mapping or function is the set of possible input
values (values of x ).
4 The range of a mapping or function is the set of output values.
5 A composite function is obtained when one function (say g) is applied
after another (say f). The notation used is g[f( x )] or gf( x ).
6 For any one-to-one function f( x ), there is an inverse function f
−
( x ).
7 The curves of a function and its inverse are reflections of each other in
the line y = x.
Functions
P
1
