(2) ( g o g ) : R ® R is given as :
( g o g ) ( x ) = g [ g (x) ] = g [ x^{2} + 1 ] = (x^{2}+1)2^{ }+ 1 = x^{4} + 2x^{2} + 2
(3) ( g o f ) : R ® R is given as :
( g o f ) ( x ) = g [ f (x) ] = g [ x^{3}] = (x^{3})^{ 2 }+ 1 = x^{6} + 1
(4) ( f o g ) : R ® R is given as :
( f o g ) ( x ) = f [ g (x) ] = f [ x^{2} + 1 ] = (x^{2}+1)^{3}
Example 3
If f(x) = log x and f (x) = x^{3}
, show that
f [ f (2) ] = 3 f (2)
Solution :
L. H. S.
f [ f (x) ] = f [ x^{3} ] = log ( x^{3} ) = 3 log x
Put x = 2 then f [ f (2) ] = 3 log 2
R. H. S.
f ( x ) = log x Þ f ( 2 ) = log 2
\ 3 f (2) = 3 log 2
\ L.H.S. = R.H.S.
Example 4
If f (x) = x^{7}  5x^{5} + 3 sin x show
that
f (x) + f (x) = 0
Solution :
f (x) = x^{7}  5x^{5} + 3 sin x is an odd function
\ f (x) = f (x) =  [ x^{7}  5x^{5}
+ 3 sin x ]
\ f (x) + f (x) = 0
Example 5
If f (x) = ( Öx )x show
that
Solution :
f (x) = ( Öx )^{ }x^{ }Þ
