Download Functions - Kineton Maths Department

Introduction
• This chapter focuses on functions
• You will learn how to answer questions
involving single and multiple functions
• You will also see how to decide whether
an expression is a function or not
Functions
Mapping Diagrams
A mapping diagram transforms one
set of numbers into a different set
of numbers. It can be described in
words or using algebra. They can
also be represented by a Cartesian
graph.
Add 3 onto the set {-3, 1, 4, 6, x)
Set A
Set B
-3
1
4
6
x
0
4
7
9
x+3
y=x+3
y
The original numbers (Set A, or ‘x’)
are known as the domain.
8
6
The results (Set B, or ‘y’) are known
as the range (ie range of answers)
4
2
-2
2
4
6
x
2A
Functions
Mapping Diagrams
A mapping diagram transforms one
set of numbers into a different set
of numbers. It can be described in
words or using algebra. They can
also be represented by a Cartesian
graph.
Square the set {-1, 1, -2, 2, x)
Set A
Set B
-1
1
-2
2
x
1
4
x2
y
The original numbers (Set A, or ‘x’)
are known as the domain.
y = x2
8
6
The results (Set B, or ‘y’) are known
as the range (ie range of answers)
4
2
-2
2
4
6
x
2A
Functions
Functions
A function is a mapping
whereby every element in
the domain is mapped to
only 1 element in the
range.
ie) Whatever number you
start with, there is only 1
possible answer to the
operation performed on it.
An example of a mapping
which is not a function
would be square rooting,
where the starting number
may result in no answer, or
2 answers.
One-to-one Function
eg) f(x) = x + 5
Set A
Set B
eg) f(x) = 3x - 2
Many-to-one Function
Set A
Set B
eg) f(x) = x2 + 1
eg) f(x) = 6 - 3x2
Not a function
Set A
Set B
eg) f(x) = √x
eg) f(x) = 1/x
2B
Functions
Functions
A function is a mapping
whereby every element in
the domain is mapped to
only 1 element in the
range.
One-to-one Function
‘A value in the
domain (x) gets
mapped to one
value in the range’
Many-to-one Function
ie) Whatever number you
start with, there is only 1 ‘Multiple values in
the domain (x) get
possible answer to the
operation performed on
mapped to the same
it.
value in the range’
An example of a mapping
which is not a function
would be square rooting,
where the starting
number may result in no
answer, or 2 answers.
‘A value in the range
can be mapped to
none, one or more
values in the range’
Not a function
2B
Functions
Example Question
Given that the function
g(x) = 2x2 + 3, find;
g(x) = 2x2 + 3
a) g(3) = 2(3)2 + 3
= 2(9) + 3
= 21
a) the value of g(3)
b) the value(s) of a such
that g(a) = 35
b)
g(a) = 35
2a2 + 3 = 35
2a2 = 32
a2 = 16
a =±4
2B
Functions
Example Question
Given that the function
g(x) = 2x2 + 3, find;
a) the value of g(3)
b) the value(s) of a such
that g(a) = 35
c) the range of the
function
 g(x) ≥ 3
y = 2x2 + 3
g(x)
You can get
any value
bigger than,
or including
3…
8
6
4
2
-2
-1
1
2
x
To work out the range of the function;
- Sketch it first
- the range is the set of answers you get (ie the
‘y’ values – now labelled as g(x)…)
- Use an Inequality if there is a continuous set
of values
2B
Functions
g(x)
x
An important bit of
notation to remember…
x can be any ‘real number’
 This is for the domain
g(x) can be any ‘real number’
 This is for the range
Real Number: A number which has a place on a
normal number line. Includes positives,
negatives, roots, pi etc…
 Does not include imaginary numbers – eg √-1
2B
Real Number: A number which
has a place on a normal number
line. Includes positives,
negatives, roots, pi etc…
Functions
Domain changes
y
A mapping which is not a function,
can be made into one by
changing/restricting the domain
(the starting values)
x
eg) y = +√x
This will not be a function as some
values in the domain (x) will not give an
answer in the range (y). For example, -2
If we restrict the domain to
x ≥ 0, then all values in the domain
will map to one value in the range.
f ( x)   x
x  R, x  0
The
function
x is real
numbers
x is
greater
than 0
 It now therefore meets the
criteria for being a function!
2C
Functions
Find the range of the following
function, and state if it is one-toone or many-to-one.
f(x) = 3x – 2, domain {x = 1, 2, 3, 4}
f(x) = 3x – 2, {x = 1, 2, 3, 4}
Domain
Range
1
1
2
4
3
7
4
10
No inequality used
as there are only
certain values
(discrete)
Range of f(x): {1, 4, 7, 10}
Description:
One to One
2C
Functions
g(x)
Find the range of the following
function, and state if it is one-toone or many-to-one.
20
15
Range
g(x) = x2, domain {x є R, -5 ≤ x ≤ 5}
g(x) = x2
10
5
g(x) = x2, {-5 ≤ x ≤ 5}
Inequality, so you will have to
sketch the graph
Range of g(x): 0 ≤ g(x) ≤ 25
Description:
Many to one
-4
-2
2
4
x
Inequality used as the
data is continuous
2C
Functions
h(x)
Find the range of the following
function, and state if it is one-toone or many-to-one.
6
Range
h(x) = 1/x, domain {x є R, 0 < x ≤ 3}
8
4
2
h(x) = 1/x, {x є R, 0 < x ≤ 3}
Inequality, so you will have to
sketch the graph
Range of h(x): h(x) ≥ 1/3
Description:
One to One
-3
-2
-1
1
2
h(x) = 1/x
x
3
In this domain, the
smallest value is 1/3
As we get close to 0, values
will get infinitely high
2C
Functions
You will need to be able to plot
more than one function on the
same set of axes, possibly for
different domains.
f(x) = 5 – 2x
8
6
4
The function f(x) is defined by:
f(x) =
{
5 – 2x
x<1
x2 + 3
x≥1
a) Sketch f(x) stating its range
f(x) > 3
b) Find the values of a such that
f(a) = 19
f(x) = x2 + 3
f(x)
2
-3
-2
-1
1
2
3
x
 Sketch both graphs on the same axes
 Make sure you use the correct domain for
each
The lowest value plotted is 3.
 Careful though as for 5 – 2x, x cannot
include 1. Therefore f(x) > 3 (not including 3)
2C
Functions
You will need to be able to plot
more than one function on the
same set of axes, possibly for
different domains.
8
6
The function f(x) is defined by:
f(x) =
{
5 – 2x
4
2
x<1
x2 + 3 x ≥ 1
a) Sketch f(x) stating its range
f(x) > 3
b) Find the values of a such that
f(a) = 19
 Solve both equations
separately!
 Remember that the answers
must be within the domain given,
or they cannot be included
f(x) = x2 + 3
y
f(x) = 5 – 2x
-3
-2
-1
1
2
Linear
Equation
5 – 2x = 19
Quadratic
Equation
x2 + 3 = 19
– 2x = 14
x2 = 16
x = -7
x = ±4
x=4
3
x
(Has to be
greater
than 1)
2C
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
Given:
f(x) = x2
g(x) = x + 1
Find:
a) fg(x)
b) gf(x)
It helps to write
what you would do
to x for each
function
f(x) = x2
g(x) = x + 1
‘Square x’
‘Add 1 to x’
a) fg(x) means g acts first, followed by f.
fg(x)
f(x + 1)
(x + 1)2
fg(x) = x2 + 2x + 1
Replace g(x) with
the function
f(x) means ‘square
x’, so square g(x)
Multiply out and
simplify
2D
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
Given:
f(x) = x2
g(x) = x + 1
Find:
a) fg(x) = x2 + 2x + 1
b) gf(x)
It helps to write
what you would do
to x for each
function
f(x) = x2
g(x) = x + 1
‘Square x’
‘Add 1 to x’
b) gf(x) means f acts first, followed by g.
gf(x)
g(x2)
(x2) + 1
Replace f(x) with
the function
g(x) means ‘add one
to x’, so add 1 to
f(x)
Simplify
gf(x) = x2 + 1
2D
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
It helps to write
what you would do
to x for each
function
Given:
f(x) = 3x + 2
g(x) = x2 + 4
Find:
a) fg(x)
b) gf(x)
c) f2(x)
d) The values of b so that fg(b) =
62
f(x) = 3x + 2
g(x) = x2 + 4
‘Multiply by 3,
then add 2’
‘Square x
then add 4’
a) fg(x) means g acts first, followed by f.
fg(x)
f(x2 + 4)
3(x2 + 4) + 2
fg(x) = 3x2 + 12 + 2
fg(x) = 3x2 + 14
Replace g(x) with
the function
f(x) means ‘multiply
by 3, then add 2’
Multiply out and
simplify
2D
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
It helps to write
what you would do
to x for each
function
Given:
f(x) = 3x + 2
g(x) = x2 + 4
Find:
a) fg(x) = 3x2 + 14
b) gf(x)
c) f2(x)
d) The values of b so that fg(b) =
62
f(x) = 3x + 2
g(x) = x2 + 4
‘Multiply by 3,
then add 2’
‘Square x
then add 4’
b) gf(x) means f acts first, followed by g.
gf(x)
g(3x + 2)
(3x + 2)2 + 4
gf(x) = 9x2 + 12x + 4 + 4
gf(x) = 9x2 + 12x + 8
Replace f(x) with
the function
g(x) means ‘square
then add 4’
Multiply out and
simplify
2D
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
It helps to write
what you would do
to x for each
function
f(x) = 3x + 2
g(x) = x2 + 4
‘Multiply by 3,
then add 2’
‘Square x
then add 4’
c) f2(x) means f acts again on itself
Given:
f(x) = 3x + 2
g(x) = x2 + 4
Find:
3x2
a) fg(x) =
+ 14
b) gf(x) = 9x2 + 12x + 8
c) f2(x)
d) The values of b so that fg(b) =
62
f2(x)
f(3x + 2)
3(3x + 2) + 2
f2(x) = 9x + 6 + 2
f2(x) = 9x + 8
Replace f(x) with
the function
f(x) means ‘multiply
by 3, then add 2’
Multiply out and
simplify
2D
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
It helps to write
what you would do
to x for each
function
Given:
f(x) = 3x + 2
g(x) = x2 + 4
Find:
a) fg(x) = 3x2 + 14
b) gf(x) = 9x2 + 12x + 8
c) f2(x) = 9x + 8
d) The values of b so that fg(b) =
62
f(x) = 3x + 2
g(x) = x2 + 4
‘Multiply by 3,
then add 2’
‘Square x
then add 4’
d) fg(b) = 62, find b
fg(b) = 62
3x2 + 14 = 62
3x2 = 48
x2 = 16
x=±4
Replace fg(b) with
the function fg(x)
Work through and
solve the equation
Remember 2
possible values
2D
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
Given:
You need to work out what order of m, n and p will
give the result when they are combined.
The best way is to do some ‘trial and error’
mentally. Looking at what is in the equation helps
speed this up.
1/
x
 2/ x
 n(x) is the only
function that contains
doubling
1/
x
m(x) =
n(x) = 2x + 4
p(x) = x2 – 2
Find in terms of m, n and p, the
function:
a) 2/x + 4
m(x) has been doubled
nm(x)
n(1/x)
2(1/x) + 4
2/ + 4
x
 so we need m(x) to
begin with, followed by
n(x)
Replace m(x)
n(x) doubles and adds 4
Multiply out and simplify
2D
Functions
Combining Functions
Two or more functions can be
combined to make a more
complex function.
Given:
m(x) = 1/x
n(x) = 2x + 4
p(x) = x2 – 2
Find in terms of m, n and p, the
function:
b) 4x2 + 16x + 14
You need to work out what order of m, n and p will
give the result when they are combined.
The best way is to do some ‘trial and error’
mentally. Looking at what is in the equation helps
speed this up.
There is a x2 in the final answer, and no
fraction, so most likely n and p are involved
If we had the x2 part first, it would only get
multiplied by 2, not 4. Whereas if we have ‘2x’
and square it, we get 4x2. Therefore n must
come before p.
pn(x)
p(2x + 4)
(2x + 4)2 - 2
4x2 + 16x + 14
Replace n(x)
p(x) squares and subtracts 2
Multiply out and simplify
2D
Functions
Inverse Functions
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
Some simple inverses
Function
f(x) = x + 4
g(x) = 2x
h(x) = 4x + 2
Inverse
f-1(x) = x - 4
g-1(x) = x/2
h-1(x) = x – 2/4
2E
Functions
Inverse Functions
Find the inverse of the
following function
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
To calculate the inverse of a
function, you need to make ‘x’ the
subject
f(x) = 3x2 - 4
y = 3x2 - 4
y + 4 = 3x2
y + 4/
3
= x2
√(y + 4/3) = x
The inverse is written
‘in terms of x’
+4
÷3
Square
root
f-1(x) = √(x + 4/3)
2E
Functions
Find the inverse of the
following function
Inverse Functions
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
To calculate the inverse of a
function, you need to make ‘x’ the
subject
m(x) =
y =
3/
(x – 1)
y(x – 1) = 3
yx - y = 3
yx = 3 + y
The inverse is written
‘in terms of x’
x=
3 + y/
y
3/
(x – 1)
Multiply by
(x – 1)
Multiply the
bracket
Add y
Divide
by y
m-1(x) = 3 + x/x
2E
Functions
Inverse Functions
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
‘The function f(x) is defined by
f(x) = √(x – 2), x ε R, x ≥ 2’
‘Find f-1(x), stating its domain.
Sketch the graphs and describe
the link between them.
Finding f-1(x)
f(x) = √(x – 2)
y = √(x – 2)
y2 = x - 2
Square
Add 2
y2 + 2 = x
f-1(x) = x2 + 2
The inverse is written
‘in terms of x’
2E
Functions
Finding the domain
of f-1(x)
Inverse Functions
‘The domain and range of a function
switch around for its inverse’
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse is
1(x)
f-
f(x)
f(x) = √(x – 2)
‘The function f(x) is defined by f(x) =
√(x – 2), x ε R, x ≥ 2’
x
‘Find f-1(x), stating its domain. Sketch
the graphs and describe the link
between them.
f-1(x) = x2 + 2
Range for f(x)  f(x) ≥ 0
Domain for f-1(x)  x ≥ 0
2E
Functions
Sketching the graph of f-1(x)
Inverse Functions
Domain is x ≥ 0, so we can draw
the graph for any values of x in
this range
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse is
1(x)
f-
f(x)
f-1(x) = x2 + 2
‘The function f(x) is defined by f(x) =
√(x – 2), x ε R, x ≥ 2’
f-1(x),
‘Find
stating its domain. Sketch
the graphs and describe the link
between them.
f(x) = √(x – 2)
x
f-1(x) = x2 + 2, {x ε R, x ≥ 0}
The link is that f(x) is
reflected in the line y = x
2E
Functions
Inverse Functions
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
If g(x) is defined as:
g(x) = 2x - 4
Domain  x ≥ 0
g-1(x) =
x + 4/
2
Domain  x ≥ -4
Range  g(x) ≥ -4 Range  g-1(x) ≥ 0
f(x)
g(x)
g-1(x)
g(x) = 2x – 4, {x ε R, x ≥ 0},
x
Calculate and sketch g(x) and g-1(x),
stating the domain of g-1(x).
Summary
• We have learnt about functions
• We have seen what is a function, and
what isn’t
• We have also learnt how to calculate
more complex functions, as well as the
inverse function