Consider the function ‘f’ exhibited by the adjoining arrow diagram.

Let A = { 1, 2, 3, 4, 5 }

B = { 5, 7, 9, 11, 13, 15 }

Note that, if we take any element x Î A then the element of the co-domain set B related to x is obtained by adding 3 to twice of x.

Applying this rule we get,

f (1) = 2 (1) + 3 = 5, f (2) = 2 (2) + 3 = 7,
f (3) = 2 (3) + 3 = 9 etc.

In general we can write f (x) = 2x + 3 x Î A

This is the formula which exhibits the function ‘f’ . If we denote the value of f at x by, y Î B, we get y = 2x + 3 x Î A. This becomes an equation which exhibits the function f .

Remarks :

1. If a function is exhibited by a formula, then using this formula we can find the range of the function.

2. Sometimes, the domain is not known or is not given. In such a case, the domain is taken as that set of elements at which the values of the function can be found. For example

   1. f(x) = x² - 3x + 4. In this case, the set R of all reals is taken as the domain of ‘f’.

   2. f(x) = In order to have a real valued function, 16 - x² ³ 0 i.e. - 4 £ x £ 4 which the desired domain of ‘f ’

   3. f (x) = then for x = 4, x + 7 is not defined. Hence in this case, the domain is the x - 4 set R of reals except 4 i.e. R - {4 }

Example 1 If A is the area of the circle. Describe this function.

Solution : If we call this function f then r is the independent variable and A becomes the dependent variable and we have A = f(r) = pr² for r ³ 0

The domain must be stated with the constraint r ³ 0 as we can’t have a circle with negative radius.

Index

Introduction

1.1 Functions And Mapping
1.2 Functions, Their Graphs and Classification
1.3 Rules for Drawing the Graph of a Curve
1.4 Classification of Functions
1.5 Standard Forms for the equation of a straight line
1.6 Circular Function and Trigonometry

Chapter 2