Question

For each pair of function f and g below, find f(g(x)) and g(f(x)).

a. f(x)=3/x,x=/= 0
b. g(x)=3/x, x=/=0
c. f(x)=x+4
d. g(x)=-x+4

Answer 1

(a) and (b)

`f(g(x)) = g(f(x)) = x`

(c) and (d)

`f(g(x)) =-x + 8`

`g(f(x)) = x + 8`

Explanation:

Given

(a) to (d)

Required

Find f(g(x)) and g(f(x)) for each pair

For (a) and (b), we have:

`f(x) = (3)/(x)`

`g(x) = (3)/(x)`

Calculate f(g(x))

`f(x) = (3)/(x)`

`f(g(x)) = (3)/(g(x))`

Substitute 3/x for g(x)

`f(g(x)) = (3)/(3/x)`

Rewrite as:

`f(g(x)) = 3/(3)/(x)`

`f(g(x)) = 3*(x)/(3)`

`f(g(x)) = x`

Since f(x) = g(x), then:

`f(g(x)) = g(f(x)) = x`

For (c) and (d)

`f(x) =x + 4`

`g(x) =-x + 4`

Solving f(g(x)), we have:

`f(x) =x + 4`

`f(g(x)) =g(x) + 4`

Substitute
`g(x) =-x + 4`

`f(g(x)) =-x + 4 + 4`

`f(g(x)) =-x + 8`

Calculating g(f(x))

`g(x) =-x + 4`

`g(f(x)) = -f(x) + 4`

Substitute:
`f(x) =x + 4`

`g(f(x)) = x + 4 + 4`

`g(f(x)) = x + 8`