,
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 '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   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)

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

R+ the set of positive real numbers

function one-one

 

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

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:

find g(x)

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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