Answer :
To find [tex]\((f \circ g)(x)\)[/tex], we need to evaluate [tex]\(f(g(x))\)[/tex].
1. Start with the function definitions:
- [tex]\(f(x) = (x + 1)^2\)[/tex]
- [tex]\(g(x) = 2x^3\)[/tex]
2. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- Since [tex]\(g(x) = 2x^3\)[/tex], we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- [tex]\(f(g(x)) = f(2x^3)\)[/tex]
3. Substitute inside [tex]\(f(x)\)[/tex]:
- Replace [tex]\(x\)[/tex] in [tex]\(f(x) = (x + 1)^2\)[/tex] with [tex]\(2x^3\)[/tex]:
- [tex]\(f(g(x)) = (2x^3 + 1)^2\)[/tex]
4. Expand the expression:
- We need to expand [tex]\((2x^3 + 1)^2\)[/tex]:
- Use the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:
- Here, [tex]\(a = 2x^3\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[
(2x^3 + 1)^2 = (2x^3)^2 + 2(2x^3)(1) + 1^2
\][/tex]
- This results in:
[tex]\[
= 4x^6 + 4x^3 + 1
\][/tex]
Thus, the expression for [tex]\(f(g(x))\)[/tex] is [tex]\(4x^6 + 4x^3 + 1\)[/tex]. So, the correct answer is [tex]\(\boxed{4x^6 + 4x^3 + 1}\)[/tex].
1. Start with the function definitions:
- [tex]\(f(x) = (x + 1)^2\)[/tex]
- [tex]\(g(x) = 2x^3\)[/tex]
2. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- Since [tex]\(g(x) = 2x^3\)[/tex], we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- [tex]\(f(g(x)) = f(2x^3)\)[/tex]
3. Substitute inside [tex]\(f(x)\)[/tex]:
- Replace [tex]\(x\)[/tex] in [tex]\(f(x) = (x + 1)^2\)[/tex] with [tex]\(2x^3\)[/tex]:
- [tex]\(f(g(x)) = (2x^3 + 1)^2\)[/tex]
4. Expand the expression:
- We need to expand [tex]\((2x^3 + 1)^2\)[/tex]:
- Use the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:
- Here, [tex]\(a = 2x^3\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[
(2x^3 + 1)^2 = (2x^3)^2 + 2(2x^3)(1) + 1^2
\][/tex]
- This results in:
[tex]\[
= 4x^6 + 4x^3 + 1
\][/tex]
Thus, the expression for [tex]\(f(g(x))\)[/tex] is [tex]\(4x^6 + 4x^3 + 1\)[/tex]. So, the correct answer is [tex]\(\boxed{4x^6 + 4x^3 + 1}\)[/tex].
