To solve the problem of finding [tex]\((h \circ g) \circ f \)[/tex], we should first understand the composition of these functions.
Given:
- [tex]\( f(x+1) = x + 3 \)[/tex]
- [tex]\( g(x+1) = x + 1 \)[/tex]
- [tex]\( h(x) = x^2 \)[/tex]
Let's break this down step-by-step:
1. Determine [tex]\( f(x) \)[/tex]:
Given [tex]\( f(x+1) = x + 3 \)[/tex], we want to find [tex]\( f(x) \)[/tex].
Substitute [tex]\( x = y - 1 \)[/tex] (so [tex]\( x + 1 = y \)[/tex]):
[tex]\[
f(y) = (y - 1) + 3 = y + 2
\][/tex]
Therefore:
[tex]\[
f(x) = x + 2
\][/tex]
2. Determine [tex]\( g(x) \)[/tex]:
Given [tex]\( g(x+1) = x + 1 \)[/tex], again we substitute [tex]\( x = y - 1 \)[/tex]:
[tex]\[
g(y) = (y - 1) + 1 = y
\][/tex]
Therefore:
[tex]\[
g(x) = x
\][/tex]
3. Determine the composition [tex]\( (g \circ f) \)[/tex]:
We know [tex]\( g(x) = x \)[/tex], so applying [tex]\( g \)[/tex] after [tex]\( f \)[/tex] we get:
[tex]\[
(g \circ f)(x) = g(f(x)) = g(x + 2)
\][/tex]
Since [tex]\( g(x) = x \)[/tex], this becomes:
[tex]\[
g(x + 2) = x + 2
\][/tex]
Therefore:
[tex]\[
(g \circ f)(x) = x + 2
\][/tex]
4. Determine the full composition [tex]\( (h \circ g) \circ f \)[/tex]:
From the previous step, we know that [tex]\( (g \circ f)(x) = x + 2 \)[/tex]. Now we need to apply [tex]\( h \)[/tex]:
We have:
[tex]\[
h(x) = x^2
\][/tex]
Therefore:
[tex]\[
h((g \circ f)(x)) = h(x + 2)
\][/tex]
Substituting into [tex]\( h \)[/tex] we get:
[tex]\[
h(x + 2) = (x + 2)^2
\][/tex]
Completing the square:
[tex]\[
(x + 2)^2 = x^2 + 4x + 4
\][/tex]
Thus, the composition [tex]\((h \circ g) \circ f\)[/tex] results in:
[tex]\[
(h \circ g) \circ f (x) = x^2 + 4x + 4
\][/tex]
