STPM
Mathematics
T / A Level
Mark Lim
STPM Mathematics T / A Level
Functions
Mark Lim
August 19, 2012
Mark Lim
STPM Mathematics T / A Level
STPM
Mathematics
T / A Level
http://marklim77.wordpress.com/author/marklim77/
Mark Lim
Mark Lim
STPM Mathematics T / A Level
What is a Function
STPM
Mathematics
T / A Level
Mark Lim
Definition
A function is a rule that produces a correspondence between
two sets of elements such that to each element in the first set
there is a corresponding one and only one element in the
second set.
Domain is a set of inputs.
Range is the set of outputs.
Mark Lim
STPM Mathematics T / A Level
Concept of Mapping
STPM
Mathematics
T / A Level
Mark Lim
1
If we enter 2 on a calculator, and the x 2 button is pressed,
then the display shows 4.
2
We can say 2 is mapped to 22
3
In other words 2 7→ 22
Mark Lim
STPM Mathematics T / A Level
Function notation
STPM
Mathematics
T / A Level
Mark Lim
Definition
If f represent ’function’ and the symbol : to mean ’such that’
and 7→ to mean ’maps to’
Then f : x 7→ x 2 , so when x = 3, x→9
Then f : x 7→ x 2 is read as ’f is the function such that x maps
to x 2
The notation f (x) is used to represent the output of the
function.It is read as ’function of x’
f:x 7→ x 2 , we now simply write f(x) = x2
Mark Lim
STPM Mathematics T / A Level
Domain and Range Illustrated
STPM
Mathematics
T / A Level
Mark Lim
Domain
Range
Domain is input and Range is the output.
Elements in Domain are mapped to elements in Range
The set X is called the domain and the set Y is called
the range of f.
Mark Lim
STPM Mathematics T / A Level
One-to-One Relations
STPM
Mathematics
T / A Level
c
d
a
b
e
f
Mark Lim
We can see the relationship from the elements in domain
and range. Arrow represents relationship.
a is mapped to b and c is mapped to d.Also e is mapped
to f .
Mark Lim
STPM Mathematics T / A Level
STPM
Mathematics
T / A Level
Mark Lim
The set of y of f(x) is called the image of the corresponding x.
Mark Lim
STPM Mathematics T / A Level
One-to-Many Relations
STPM
Mathematics
T / A Level
Mark Lim
c
d
a
b
e
f
We can see a is mapped to b and d.
It is known as one-to-many relations.
Mark Lim
STPM Mathematics T / A Level
Example f : x 7→ 2x
STPM
Mathematics
T / A Level
Mark Lim
f
3
6
f
2
4
f
10
20
When the value of the domain input, x is 10, its output
range is 20 as governed by the function and so on
The governing function stated explicitly as f : x 7→ 2x
Mark Lim
STPM Mathematics T / A Level
Composite Functions
STPM
Mathematics
T / A Level
Mark Lim
X
Y
Z
g
f
gf
The function f maps from set X to set Y.
The function g maps from set Y to set Z.
Composite function gf now maps from X to Z.
Mark Lim
STPM Mathematics T / A Level
Usage of the composite notation ◦
STPM
Mathematics
T / A Level
Mark Lim
We are using the f and g as the two functions.
1
g ◦ f is known as composition of g of f.
It means g f(x)
2
f ◦ g means fg(x)
Note the difference between the two types of notation carefully.
Mark Lim
STPM Mathematics T / A Level
Example of Composite gf
STPM
Mathematics
T / A Level
Mark Lim
If f : x 7→ 2x 2 + 1 and g : x 7→ x + 3 , find composite gf(x)
Here we go for the function f first.
f (x) = 2x 2 + 1
g (x) = x + 3
gf (x) = g [(2x 2 + 1) + 3]
∴ gf (x) = 2x 2 + 4
Mark Lim
STPM Mathematics T / A Level
Example of Composite fg
STPM
Mathematics
T / A Level
Mark Lim
If f : x 7→ 2x 2 + 1 and g : x 7→ x + 3, find composite fg
Here we go for g function first.
fg (x) = f (x + 3)
fg (x) = 2(x + 3)2 + 1
fg (x) = 2(x 2 + 6x + 9) + 1
= 2x 2 + 12x + 18 + 1
= 2x 2 + 12x + 19
Notice the major difference between gf(x) and fg(x).
Mark Lim
STPM Mathematics T / A Level