If  f(x) = 3x^5  and  g(x)= \frac{x-1}{2x^2}  , then what is the composite function of f(g(x)).

Question 20Answer

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

b.
\frac{3x^5-1}{18x^{10}}

c.
\frac{3x^5-1}{18x^7}

d.
\frac{3x^5-1}{6x^7}

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.

Understand how to find the composite function f(g(x)) with a step-by-step explanation. This problem involves function composition, using the functions f(x) = 3x^5 and g(x) = (x-1)/(2x^2). Learn the method to substitute g(x) into f(x), and solve for the correct expression. Suitable for high school students in Grades 10-12.

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