Please show me how to solve this problem:

Question

Let f(x)=x^2 and g(x)=x+4 and find
a. (fog)(5)
b. (gof)(5)
c. (fog)(x)
d. (gof)(x)
These problems are called the composition of f with g and the composition of g with f. Equations and inequalities in two variables.

Answer 1

Given functions are f (x ) = x ^2  and g (x ) = x + 4

a) ( f o g )(5 ) = f ( g (5 ))

Substitute the expression for functioning g  (in this case x +4 ) for g (x ) in the composition.

g (5 ) = 5 + 4 = 9

f ( g (5 )) = f ( 9 )

Substitute the expression for functioning f  (in this case x ^2 ) for f (x ) in the composition.

 = 9^2 = 81

( f o g )(5 )  = 81.

b) ( g o f )(5 ) = g ( f (5 ))

Substitute the expression for functioning f  (in this case x ^2 ) for f (x ) in the composition.

f ( 5 ) = 5^2 = 25

g ( f (5 )) = g (25)

Substitute the expression for functioning g  (in this case x + 4 ) for g (x ) in the composition.

( g o f )(5 ) = 25 + 4 = 29.

c) ( f o g )(x ) = f ( g (x ))

Substitute the expression for functioning g  (in this case x + 4 ) for g (x ) in the composition.

= f (x + 4)

Now substitute this expression (x + 4) in to function f  in place of the x  value.

= (x + 4)^2

( f o g )(x ) = x ^2 + 16 + 8x.

d) ( g o f  )(x ) = g ( f (x ))

Substitute the expression for functioning f  (in this case x ^2 ) for f (x ) in the composition.

= g (x ^2)

Now substitute this expression (x ^2 ) in to function g  in place of the x  value.

= x ^2  + 4

( g o f  )(x ) =  x ^2 +4.