Answer :
To find the expression for [tex]\((f^{\circ} g)(x)\)[/tex], which is the composition of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we need to follow these steps:
1. Identify the Given Functions:
- [tex]\(f(x) = 6x + 9\)[/tex]
- [tex]\(g(x) = x^4\)[/tex]
2. Understand Function Composition:
- The notation [tex]\((f^{\circ} g)(x)\)[/tex] represents the composition of [tex]\(f\)[/tex] and [tex]\(g\)[/tex].
- This means we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]: [tex]\(f(g(x))\)[/tex].
3. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- Since [tex]\(g(x) = x^4\)[/tex], substitute this into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(x)) = f(x^4)
\][/tex]
4. Evaluate [tex]\(f(x^4)\)[/tex]:
- Replace [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex] with [tex]\(x^4\)[/tex]:
[tex]\[
f(x^4) = 6(x^4) + 9
\][/tex]
5. Simplify the Expression:
- Simplify to get the expression for [tex]\((f^{\circ} g)(x)\)[/tex]:
[tex]\[
f(g(x)) = 6x^4 + 9
\][/tex]
Thus, the expression for [tex]\((f^{\circ} g)(x)\)[/tex] is [tex]\(6x^4 + 9\)[/tex].
The correct choice from the given options is:
- [tex]\(6x^4 + 9\)[/tex]
1. Identify the Given Functions:
- [tex]\(f(x) = 6x + 9\)[/tex]
- [tex]\(g(x) = x^4\)[/tex]
2. Understand Function Composition:
- The notation [tex]\((f^{\circ} g)(x)\)[/tex] represents the composition of [tex]\(f\)[/tex] and [tex]\(g\)[/tex].
- This means we substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]: [tex]\(f(g(x))\)[/tex].
3. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
- Since [tex]\(g(x) = x^4\)[/tex], substitute this into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(x)) = f(x^4)
\][/tex]
4. Evaluate [tex]\(f(x^4)\)[/tex]:
- Replace [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex] with [tex]\(x^4\)[/tex]:
[tex]\[
f(x^4) = 6(x^4) + 9
\][/tex]
5. Simplify the Expression:
- Simplify to get the expression for [tex]\((f^{\circ} g)(x)\)[/tex]:
[tex]\[
f(g(x)) = 6x^4 + 9
\][/tex]
Thus, the expression for [tex]\((f^{\circ} g)(x)\)[/tex] is [tex]\(6x^4 + 9\)[/tex].
The correct choice from the given options is:
- [tex]\(6x^4 + 9\)[/tex]
