1. A vertical line has equation x = a ( y-axis has equation
   x = 0 ) , a Î R

2. A horizontal line has equation y = b (x-axis has equation y = 0), b Î R

If two straight lines are a₁x + b₁y + c₁ = 0 and a₂ x + b₂y + c₂ = 0

then ¹          for parallel lines

and (a₁a₂ + b₁b₂ = 0 )      for mutually perpendicular lines.

Example 1 If f : R ® R. f(x) = 4x-3 x Î R. Find f ^-1, also find f o f ^-1 and f ^-1 o f.

Solution :

Let y = f ( x ) = 4x-3 then 4x = y+3 \ x =

\ f ^-1 : R ® R is given by f ^-1 ( y ) =

or f ^-1 R ® R is given by f ^-1 (x) = (replacing y by x)

\ (f ^-1 o f ) : R ® R is given by

( f ^-1 o f ) ( x ) = f ^-1 [ f(x) ] = f ^-1 ( 4x-3 ) = = x

and (f ^-1 o f) : R ® R is given by

Example 2

Let f : R ® R and g : R ® R and f (x) = x³ and g (x) = x² + 1.

Find (1) f o f (2) g o g (3) g o f (4) f o g

(1) ( f o f ) : R ® R is given as :

( f o f ) ( x ) = f [ f (x) ] = f [x³] = (x³) ³ = x9

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