C3 Core Maths (Steph Richards)
MAPPINGS and FUNCTIONS
A function is a special type of mapping such that each member of
the domain is mapped to one, and only one, element in the range.
DOMAIN
The DOMAIN is the set of ALLOWED INPUTS TO A FUNCTION.
RANGE
The RANGE is the set of POSSIBLE OUTPUTS FROM A FUNCTION
RANGE
DOMAIN
RANGE
y  2x  1
DOMAIN
MANY TO ONE MAPPING
DOMAIN
RANGE
RANGE
ONE TO ONE MAPPING
y  x2
DOMAIN
The DOMAIN is usually defined as an interval (as in the graphical representation) rather than a list of
DISCRETE numbers (as in the mapping diagram)
C3 Core Maths (Steph Richards)
MANY TO MANY MAPPING
ONE TO MANY MAPPING
•
•
•
•
•
•
•
•
•
•
2
-2
0
√8
-√8
2
-2
0
√8
-√8
x2  y2  8
y x
These MAPPINGS DO NOT represent FUNCTIONS because there is
not a UNIQUE OUTPUT for a given INPUT.
CHECK:
Place the following mappings in the table:
FUNCTIONS(almost!)
One-one
mapping
Many-one
mapping
NOT FUNCTIONS
One-Many
mapping
Place the following mappings in the table below
1
y
y  3 x
(a)
(b)
(c)
y2  x  3
x
y x
y  x4
y  x3
(e)
(f)
(g)
Many-Many
mapping
(d)
x2 y2

1
4
9
(h)
y  3x
C3 Core Maths (Steph Richards)
The DOMAIN must be defined (or its not a function!)


Elements that are NOT allowed into the function are identified.
The DOMAIN may be purposly restricted for some reason. (eg to make the
function become one to one)
EXAMPLES
1
x
Will not allow the value of x=0.
y
The FUNCTION is defined as
y  x 1
Will not allow values of x less than -1.
Also it is a one to many mapping so is not a function.
The FUNCTION can be defined along with its DOMAIN:
INTERVAL NOTATION
This is a simple way to write an interval but without inequalities.
C3 Core Maths (Steph Richards)
FINDING THE RANGE
The RANGE of a function can be visualised as the projection onto the y axis
Find the RANGE of the one to one function defined as
f ( x)  x -1
2
x 1
The RANGE of the function:
This is a ONE TO ONE FUNCTION.
The RANGE can be found by
substituting (where allowed) the
end values of the DOMAIN into
the function.
C3 Core Maths (Steph Richards)
The RANGE of a “MANY TO ONE FUNCTION” will need careful consideration.
Find the RANGE of the function defined as:
f ( x )  x 2 -1
x > 2
The DOMAIN is different so the function is different.
The set of values in the RANGE written in
INTERVAL NOTATION is
For a “Many to one Function” we need to identify
stationary points, in order to find the minimum
(in this case) or Maximum values in the RANGE.
EXAMPLES
(Given that these functions are ONE TO ONE)
Write down the range of the function (using interval notation)
 2,  
C3 Core Maths (Steph Richards)
CHECK UP:
Find the RANGE of the given functions. Write the RANGE using INTERVAL
NOTATION. (TAKE CARE IF A FUNCTION IS NOT ONE TO ONE)
Function and Domain
Range
1
f  x    x , x R
1
2
f  x   2 x  10, x  5
2
3
f  x   10  2 x, x  5
3
4
f  x    x2 , x  2
4
5
f  x   2x  10, x  0
5
6
f  x   10  2x, x  2
6
7
f  x   x2  2, x  1
7
8
f  x    x2 , x  3
8
9
f  x   x2  2, x  1
9
10
f  x   x2  2, x  0
10
2
Question 7 needs care as it is NOT one to one wih the domain given.
C3 Core Maths (Steph Richards)
THE INVERSE FUNCTION
f ( x)
DOMAIN
f(x)
x maps on to y and so y maps
back onto x.
RANGE
f(x)
RANGE
THE GRAPH OF THE INVERSE
FUNCTION IS THE REFLECTION
IN THE LINE y=x OF THE
ORIGINAL FUNCTION.
DOMAIN
-1
f (x)
-1
1
f (x)
f ( x)
VICA VERSA
We think of the INVERSE FUNCTION as the function that “works backwards”
For an INVERSE to EXIST the original function MUST BE ONE TO ONE
-1
DOMAIN f(x) is EQUAL TO RANGE f (x)
-1
RANGE f(x) is EQUAL TO DOMAIN f (x)
To find the INVERSE FUNCTION we let y  f ( x ) and change the subject of
the formula.
EXAMPLE
A function is defined as
f ( x)  x  2
x2
1
Find the inverse function f ( x ) . Find the domain and Range of f 1 ( x ) and sketch the
1
graphs of f ( x ) and f ( x ) on the same pair of axes.
The INVERSE FUNCTION is
-1
DOMAIN f(x) is EQUAL TO RANGE f (x) ………………………………………..
-1
RANGE f(x) is EQUAL TO DOMAIN f (x)…………………………………………
yx
C3 Core Maths (Steph Richards)
f 1 ( x ) 
f ( x)  x  2
EXAMPLE
Given that:
x2  3
f ( x)  2
x 5
for  ,0 
Find an expression for f
f ( x )  0
1
( x)
And find the DOMAIN and RANGE of f
1
( x)
for all values of x on the domain
C3 Core Maths (Steph Richards)
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