PURE MATHEMATICS - Algebra

 

Functions

 

 

Mapping

Inverse

Composite

Exponential & Log

 

 

 

Introduction

 

To thoroughly understand the terms and symbols used in this section it is advised that you visit 'Number Set Theory ' 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   x1 x2 x3...   y1 y2 y3...

 

 

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 = x2 can be more accurately expressed using the following format:

 

set theory format for an equation

 

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).

 

 

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One-One Mapping

 

Here one element of the domain is associated with one and only one element of the range.

 

A property of one-one functions is that a on a graph a horizontal line will only cut the graph once.

 

 

Example

 

one-one mapping example

 

Where R+ is the set of positive real numbers.

 

 

function one-one

 

 

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Many-One Mapping

 

Here more than one element of the domain can be associated with one particular element of the range.

 

 

Example

 

many-one function mapping

 

Where Z is the set of integers(positive & negative whole numbers NOT including zero).

 

 

functions many-one

 

 

Complete Function Notation is a variation on what has been used so far.

 

It will be used from now on.

 

 

function notation

 

 

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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-1f(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.

 

 

inverse function problem#1

 

 

inverse function problem#1 graph

 

 

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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.

 

composite functions conditions

 

The method for finding composite functions is simply:

 

1. Find g(x).

 

2. Find f[g(x)].

 

 

Example

 

For the two functions,

 

composite function problem#1

 

 

find the composite functions    (i  fg   (ii  g f

 

 

composite functions problem#1b

 

 

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Exponential & Logarithmic Functions

 

Exponential functions have the general form:

 

an exponential function

 

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.

 

exponential function

 

The function(above) has one-one mapping.

 

It therefore possesses an inverse. This inverse is the logarithmic function.

 

 

log is inverse of exponential

 

 

inverse of exponential function

 

 

 

 

 

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