Find the best solutions to your problems with the help of IDNLearn.com. Ask your questions and receive accurate, in-depth answers from our knowledgeable community members.
Sagot :
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]
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]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.