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If [tex]$f(x)=3 x^5$[/tex] and [tex]$g(x)=\frac{x-1}{2 x^2}$[/tex], then what is the composite function [tex][tex]$g(f(x))$[/tex][/tex]?

A. [tex]\frac{3 x^5-1}{18 x^{10}}[/tex]
B. [tex]3\left(\frac{x-1}{2 x^2}\right)^5[/tex]
C. [tex]\frac{3 x^5-1}{18 x^7}[/tex]
D. [tex]\frac{3 x^5-1}{6 x^7}[/tex]



Answer :

GinnyAnswer
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]

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