From the mapping g, represented to the diagram above:

• The set of b, c, d, I is called the Domain
• The set of p, c, q, I is called the co-domain

Definition;

A function is a mapping in which every element in the domain maps onto one and only one element in the co-domain.

A function can either be a one-to-one or many-to-one mapping (diagram seen in relations above).

A function is usually denoted be f, g, h etc. if f is a function which maps elements of set A to elements of set B then, we write it as:

f : A          B

Finding the range of a function

The range of a function f is obtained by finding the images of the elements in the diagram.

Example;

Given that f : x          2x + 1, find the range of f, if the domain D= (-1, 0, 1, 2)

Solution

f:x         2x +1

f(x) = 2x+1

f(-1) = 2(-1) +1,   f(0) = 2(0) +1,   f(1) = 2(1) +1,  f(2) = 2(2)+1

        = -1                       =1                       = 3                    =5

Therefore, range = (-1, 1, 3, 5)

Any question involving range of a function can be solve similarly

Finding the domain of a function

The domain of a function is the set of elements having an image in the co-domain. The domain of a function f is often denoted be D_f.

Examples;

State the domain of the following functions:

1. f:x         2x + 3
2. g:x          1x-2

Solution

1. D_f = x Є real numbers or xЄR
2. g:x          1x-2

Let the divisor x-2=0

x = 2

Therefore, D_g = xЄR –(2)

Inverse of a function

For a given function f, which maps set A to set B (f:A           B), the inverse of the function denoted as f^-1 is that which maps set B to set A (f^-1 : B         A).

To find the inverse of the function f(x), follow the steps below;

• Let f(x) = y
• Make x the subject of the formula
• Replace x by f^-1(x) and y by x

Example;

Find the inverse of the following functions;

1. f:x          x – 2
2. g:x          x+23

Solution

1. f(x) = x – 2                                  b) g(x) = x+23

let f(x) = y                                      let g(x) = y

x – 2 = y                                         x+23=y

x = y +2                                       3y-2 = x

         f^-1 (x) = x + 2                              therefore, g^-1 (x) = 3x – 2

f of a function

f of a given function g(x) is denoted as f o g(x) or f[g(x)].

Given the function f(x) = 2x + 3, f(2)= 2(2) + 3, similarly,

 fog(x) = 2[g(x)] + 3.

f of a function means, the function in f

Example;

Given that f(x) = x +1 and g(x) = 2x -3, find;

1. f o g(x)     b) g o f(x)

Solution

1. f o g(x) = g(x) + 1                         b) g o f(x) = 2[f(x)] - 3

             = (2x-3) + 1                                        = 2(x + 1) - 3

f o g(x) = 2x -2                                  g o f(x) = 2x + 2 – 3

                                                                         = 2x – 1