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Example 2 A relation is given as : { (1, 1), (2, ½ ), (3, ) , (4, ¼) , (5, ) } is it a function ? Justify. If it is a function state its domain and range. Exhibit the relation by a formula.

Solution :

Let A = { 1, 2, 3, 4, 5} and B = {1, ½, , ¼ , }

Every element in A occurs once and only once in the ordered pairs and is related to one and only one element of the set B, exhibiting a relation. Hence it is a function from the set A to the set B. Let us denote it by f. Then domain of A = The set A and the range of f = The set B

Also, the second member in each ordered pair is the reciprocal of the first member. Thus if we denote any element of the set A by x and that of B by y then

The formula is f(x) = 1/x i.e. y = 1/x x Î A

Example 3 A function f is given as f(x) = x2 - 6x + 5. State its domain. Find f(-2), f(2), f(1), f(a + h), ,h ¹ 0

Solution :

The domain of 'f ' is the set R of all reals

f(x) = x2 - 6x + 5 Þ f (-2) = (-2) 2 - 6 (-2) + 5 = 21

f(2) = (2) 2 - 6 (2) + 5 = - 3, f (1) = (1)2 - 6 (1) + 5 = 0

f(a + h) = (a + h) 2- 6 (a + h) + 5 = a2+ 2ah + h2- 6a - 6h+ 5

and

=  


Example 4 Find the range of each of the following functions

(1) f(x) = 3x + 4 for - 5 £ x £ 8

(2) g(x) = 2x2 + 3 for - 4 £ x £ 3

(3) f (x) = x2- 6x + 7 for all x Î R

Solution :

(1)     The domain of ' f ' = { x | x Î R, - 5 £ x £ 8 }

Now -5 £ x £ 8 \ 3 (-5) £ 3 (x) £ 3 (8)

\ - 15 £ 3x £ 24 \ - 15 + 4 £ 3x + 4 £ 24 + 4

\ - 11 £ f(x) £ 28

The range of ' f ' = { f(x) Î R, - 11 £ f(x) £ 28 }

(2)    - 4 £ x £ 3 Þ 0 £ x2 £ 16 (why ? )

\ 2 (0) £ 2 (x) £ 2 (16) \ 0 + 3 £ 2 x2 + 3 £ 30 + 3

\ 3 £ 9(x) £ 35

This is the range of 'g '

 

 

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





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