Introduction
To thoroughly understand the terms and symbols used in this section it is advised that you visit 'sets of numbers' first.
Mapping(or function)
This a 'notation' for expressing a relation between two variables(say x and y).
Individual values of these variables are called elements
eg x_{1} x_{2} x_{3}... y_{1} y_{2} y_{3}...
The first set of elements ( x) is called the domain .
The second set of elements ( y) is called the range .
A simple relation like y = x^{2} can be more accurately expressed using the following format:
The last part relates to the fact that x and y are elements of the set of real numbers R(any positive or negative number, whole or otherwise, including zero)
OneOne mapping
Here one element of the domain is associated with one and only one element of the range.
A property of oneone functions is that a on a graph a horizontal line will only cut the graph once.
Example
R^{+} the set of positive real numbers
ManyOne mapping
Here more than one element of the domain can be associated with one particular element of the range.
Example
Z is the set of integers(positive & negative whole numbers not including zero)
Complete function notation is a variation on what has been used so far. It will be used from now on.
Inverse Function f ^{1}
The inverse function is obtained by interchanging x and y in the function equation and then rearranging to make y the subject.
If f ^{1} exists then,
ff^{1}(x) = f^{1}f(x) = x
It is also a condition that the two functions be 'one to one'. That is that the domain of f is identical to the range of its inverse function f ^{1} .
When graphed, the function and its inverse are reflections either side of the line y = x.
Example
Find the inverse of the function(below) and graph the function and its inverse on the same axes.
Composite Functions
A composite function is formed when two functions f, g are combined.
However it must be emphasized that the order in which the composite function is determined is important.
The method for finding composite functions is:
find g(x)
find f[g(x)]
Example
For the two functions,
find the composite functions (i fg (ii g f
Exponential & Logarithmic Functions
Exponential functions have the general form:
where 'a' is a positive constant
However there is a specific value of 'a' at (0.1) when the gradient is 1 . This value, 2.718... or 'e' is called the exponential function.
The function(above) has oneone mapping. It therefore possesses an inverse. This inverse is the logarithmic function.

