For fx2x3 and gxx21 find the composite functions defined by

For f(x)=2x+3 and g(x)=-x2+1, find the composite functions defined by

I.(fg)(x)

II.(gf)(x)

Solution

Given that

f(x) = 2x + 3 , g(x) = -x² + 1

1 ) ( f o g )(x) = f ( g(x) )

= f (  -x² + 1 )

= 2 (  -x² + 1 ) + 3 [ since ,  f(x) = 2x + 3 ]

= -2x² + 2 + 3

( f o g )(x) = -2x² + 5

2 ) ( g o f )(x) = g ( f(x) )

= g ( 2x + 3 )

= - ( 2x + 3 )² + 1 [ since ,  g(x) = -x² + 1 ]

= - ( (2x)² + 2.2x.3 + (3)² ) + 1 [ since , ( a + b )² = a² + 2ab + b² ]

= - ( 4x² + 12x + 9 ) + 1

= - 4x² - 12x - 9 + 1

( g o f )(x) = - 4x² - 12x - 8