To solve the problem of finding suitable functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] such that [tex]\( h(x) = (f \circ g)(x) = 5(x+1)^3 \)[/tex], let's break it down step-by-step.
Given:
[tex]\[ h(x) = 5(x + 1)^3 \][/tex]
We need to find functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] such that:
[tex]\[ h(x) = f(g(x)) \][/tex]
Let's examine the options provided:
### Option A:
[tex]\[ f(x) = 5 x^3 \][/tex]
[tex]\[ g(x) = (x + 1)^3 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f((x + 1)^3) = 5((x + 1)^3)^3 \][/tex]
This simplifies to:
[tex]\[ f(g(x)) = 5(x + 1)^9 \][/tex]
This does not match [tex]\( h(x) \)[/tex].
### Option B:
[tex]\[ f(x) = 5 x^3 \][/tex]
[tex]\[ g(x) = x + 1 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x + 1) = 5(x + 1)^3 \][/tex]
This simplifies to:
[tex]\[ f(g(x)) = 5(x + 1)^3 \][/tex]
This matches [tex]\( h(x) \)[/tex].
### Option C:
[tex]\[ f(x) = (x + 1)^3 \][/tex]
[tex]\[ g(x) = 5 x^3 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(5 x^3) = (5x^3 + 1)^3 \][/tex]
This does not match [tex]\( h(x) \)[/tex].
### Option D:
[tex]\[ f(x) = x + 1 \][/tex]
[tex]\[ g(x) = 5 x^3 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(5 x^3) = 5x^3 + 1 \][/tex]
This does not match [tex]\( h(x) \)[/tex].
Given the solutions above, only Option B is a valid combination of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] that satisfies [tex]\( h(x) = 5(x + 1)^3 \)[/tex].
Therefore, one possibility for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is:
[tex]\[ f(x) = 5 x^3 \][/tex]
[tex]\[ g(x) = x + 1 \][/tex]
