Higher Maths 1.2.1 - Sets and Functions

1. Higher Maths 1 2 Functions 1

2. The symbol means ‘is an element of’.
Introduction to Set Theory
In Mathematics, the word set refers to a group of numbers or other
types of elements. Sets are written as follows:
Examples
{ 1, 2, 3, 4, 5, 6 }{ -0.7, -0.2, 0.1 }{ red, green, blue }
4 { 1, 2, 3, 4, 5 }
7 { 1, 2, 3 }
{ 6, 7, 8 } { 6, 7, 8, 9 }
If A = { 0, 2, 4, 6, 8, … 20 }
and B = { 1, 2, 3, 4, 5 } then B A
Sets can also be named using letters:
P = { 2, 3, 5, 7, 11, 13, 17, 19, 23, … }
Higher Maths 1 2 1 Sets and Functions 2

3. N
W
Z
{ 1, 2, 3, 4, 5, ... }
{ 0, 1, 2, 3, 4, 5, ... }
{ ... -3, -2, -1, 0, 1, 2, 3, ... }
The Basic Number Sets
Q
Rational numbers
Includes all integers, plus any number which
can be written as a fraction.
R
√7π
Includes all rational numbers, plus irrational
numbers such as or .
Real numbers
C
Complex numbers
Includes all numbers, even imaginary ones
which do not exist.
Whole numbers
Integers
Natural numbers
NWZQRC
2
3Q√-1R
Examples
Higher Maths 1 2 1 Sets and Functions 3

4. Set Theory and Venn Diagrams
Venn Diagrams are illustrations which use overlapping
circles to display logical connections between sets.
Blue
Animal
Food
Pig
Blueberry
Pie
Blue
Whale
?
Rain
Red
Yellow
Orange
Juice
Sun
Strawberries
Aardvark
N
W
Z
Q
R
C
√82
5
7
Higher Maths 1 2 1 Sets and Functions 4

5. Function Domain and Range
Any function can be thought of as
having an input and an output.
The ‘input’ is sometimes also known
as the domain of the function, with
the output referred to as the range.
f (x)
domain range
Each number in the domain has a
unique output number in the range.
The function
has the domain
{ -2, -1, 0, 1, 2, 3 }
Find the range.
Imporant
Example
f (x) = x
2
+ 3x
f (-2) = 4 – 6 = -2
f (-1) = 1 – 3 = -2
f (0) = 0 + 0 = 0
f (1) = 1 + 3 = 4
f (2) = 4 + 6 = 10
Range = { -2, 0, 4, 10 }
Higher Maths 1 2 1 Sets and Functions 5

6. Composite Functions
It is possible to combine functions by
substituting one function into another.
f (x) g (x)
g ( )f (x)
is a composite function
and is read ‘ ’.
g ( )f (x)
g of f of x
Important
g ( )f (x)f ( )g (x)≠
In general
Given the functions
Example
g(x) = x + 3
f (x) = 2 x
and
find and .
= 2 ( )x + 3
= 2 x + 6
= ( ) + 32 x
= 2 x + 3
f ( g (x))g ( f (x))
g ( f (x))
f ( g (x))
Higher Maths 1 2 1 Sets and Functions 6

7. = x
Inverse of a Function
If a function also works backwards for
each output number, it is possible to write
the inverse of the function.
f (x)
f (x) = x
2
f (4) = 16
f (-4) = 16
Not all functions have an inverse, e.g.
Every output in the range must have only
one input in the domain.
does not have
an inverse function.
f (x) = x
2

f (16) = ?
-1
domain range
Note that
f ( )
-1
f (x)
f (x)
-1
= x
x
and
f ( )f (x)
-1
Higher Maths 1 2 1 Sets and Functions 7

8. Find the inverse function for .

Finding Inverse Functions
g (x) = 5 x
3
– 2
-1
Example
g (x)
3
× 5 – 2
g
x
x + 2
√
3
÷ 5 + 2
g-1
x
x
x + 2
5
x + 2
5
3
-1
g (x) =
+ 2
÷ 5
3
Higher Maths 1 2 1 Sets and Functions 8

9. Graphs of Inverse Functions
To sketch the graph of an inverse function , reflect
the graph of the function across the line .
f (x)
-1
f (x) y = x
y = x
f (x)
-1
f (x)
y = x
g (x)
-1
g (x)
Higher Maths 1 2 1 Sets and Functions 9

10. Basic Functions and Graphs
f (x) = ax
f (x) = a sin bx
f (x) = a tan bx
f (x) = ax²
f (x) = ax³
f (x) =
a
x
Linear Functions
Quadratic Functions
Trigonometric Functions
Cubic Functions Inverse Functions
Higher Maths 1 2 1 Sets and Functions 10

11. Exponential and Logartithmic Functions
f (x) = log xa
1
f (x) = ax
1
(1,a)
(1,a)
is called an exponential function with base .
Exponential Functions
f (x) = a
x
a
The inverse function of an exponential function is called
a logarithmic function and is written as .f (x) = log xa
Logarithmic Functions
Higher Maths 1 2 1 Sets and Functions 11

12. Finding Equations of Exponential Functions
Higher Maths 1 2 1 Sets and Functions 12
It is possible to find the equation of any
exponential function by substituting values
of and for any point on the line.
y = a + b
x
2
(3,9)
Example
The diagram shows
the graph of
y = a + b
Find the values
of a and b.
Substitute (0,2):
x
2 = a + b
0
= 1 + b
b = 1
xy
Substitute (3,9):
9 = a + 1
3
a = 2
a = 8
3
y = 2 + 1
x