Alright, let's carefully determine the composite function [tex]\(g(f(x))\)[/tex] step by step.
1. Define the given functions:
- [tex]\( f(x) = 3x^5 \)[/tex]
- [tex]\( g(x) = \frac{x - 1}{2x^2} \)[/tex]
2. Form the composite function [tex]\( g(f(x)) \)[/tex]:
- To do this, we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
3. Substitute [tex]\( f(x) = 3x^5 \)[/tex] into [tex]\( g(x) \)[/tex]:
- Substitute [tex]\( 3x^5 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(3x^5) = \frac{3x^5 - 1}{2(3x^5)^2}
\][/tex]
4. Simplify the expression inside the function [tex]\( g \)[/tex]:
- Calculate [tex]\( (3x^5)^2 \)[/tex]:
[tex]\[
(3x^5)^2 = 9x^{10}
\][/tex]
5. Substitute back into the composite function:
- Now the composite function becomes:
[tex]\[
g(f(x)) = \frac{3x^5 - 1}{2 \cdot 9x^{10}} = \frac{3x^5 - 1}{18x^{10}}
\][/tex]
6. Final Simplified Form:
- So the final simplified form of the composite function [tex]\( g(f(x)) \)[/tex] is:
[tex]\[
g(f(x)) = \frac{3x^5 - 1}{18x^{10}}
\][/tex]
Given the answer choices, we see that the correct choice is:
[tex]\[
\boxed{\frac{3x^5 - 1}{18x^{10}}}
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{a. } \frac{3x^5 - 1}{18x^{10}}
\][/tex]
