1.5 Standard Forms For The Equation Of A Straight Line

A vertical line has equation x = a ( yaxis has equation x = 0 ) , a Î R
A horizontal line has equation y = b (xaxis has equation y = 0), b Î R
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If two straight lines are a_{1}x +
b_{1}y + c_{1} = 0 and a_{2} x + b_{2}y
+ c_{2} = 0
then ¹ for parallel lines
and (a_{1}a_{2} + b_{1}b_{2}
= 0 ) for mutually perpendicular lines.
Example 1 If f : R ®
R. f(x) = 4x3 x
Î R. Find f ^{1}, also
find f o f ^{1} and f ^{1} o f.
Solution :
f : R ® R is oneone onto. Hence f ^{1}
exists and f (x) = 4x  3
Let y = f ( x ) = 4x3 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} ( 4x3 ) =
= x
and (f ^{1} o f) : R ®
R is given by
( f o f ^{1} ) ( x ) = f [ f ^{1} (x) ] =
Example 2
Let f : R ® R and g : R ®
R and f (x) = x^{3} and g (x) = x^{2} + 1.
Find (1) f o f (2) g o g (3) g o f (4) f o g
Solution :
(1) ( f o f ) : R ® R is given as :
( f o f ) ( x ) = f [ f (x) ] = f [x^{3}]
= (x^{3})^{ 3} = x9
