To find the composite function $f(g(x))$, you substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = 3x^5$
• $g(x) = \frac{x-1}{2x^2}$

Now, $f(g(x)) = f\left(\frac{x-1}{2x^2}\right)$.

Substitute $\frac{x-1}{2x^2}$ into $f(x)$:

$f(g(x)) = 3\left(\frac{x-1}{2x^2}\right)^5$

This is the expression for the composite function $f(g(x))$.

Therefore, the correct answer is:

a. $3\left(\frac{x-1}{2x^2}\right)^5$

Would you like more details or have any questions?

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Here are 8 related questions:

1. How do you find the composite of two functions?
2. What is the domain of the composite function $f(g(x))$?
3. How do you simplify expressions like $\left(\frac{x-1}{2x^2}\right)^5$?
4. Can composite functions have different domains than the original functions?
5. What happens to the range of a function when you take a composite function?
6. How do you determine if a composite function is one-to-one?
7. Can the order of composition $f(g(x))$ and $g(f(x))$ affect the outcome?
8. How would you find the inverse of the composite function $f(g(x))$?

Tip: When composing functions, always check both the domain and range of the resulting function to ensure it's valid in the given context.