https://file.aiquickdraw.com/mathbot/file/1726801810255giq72br6.jpg

You are tasked with finding two functions, \( f(x) \) and \( g(x) \), such that:

\[
f(g(x)) = \sqrt[5]{\frac{x+1}{x+9}}
\]

This is a composition of functions problem, where we need to decompose the given expression into two simpler functions \( f(x) \) and \( g(x) \). 

### Step-by-step approach:

1. **Identify the outer function \( f(x) \):**  
   Notice that the given expression involves a fifth root, which suggests that the outer function could handle this root. We can express \( f(x) \) as follows:
   \[
   f(x) = \sqrt[5]{x}
   \]
   This means that \( f(x) \) takes some input \( x \) and gives the fifth root of it.

2. **Identify the inner function \( g(x) \):**  
   Since \( f(g(x)) = \sqrt[5]{\frac{x+1}{x+9}} \), the inner function must handle the fraction inside the root. Thus, we set \( g(x) \) to be:
   \[
   g(x) = \frac{x+1}{x+9}
   \]

### Conclusion:

The two nontrivial functions are:
- \( f(x) = \sqrt[5]{x} \)
- \( g(x) = \frac{x+1}{x+9} \)

These functions satisfy the given condition for the composition \( f(g(x)) \).

Would you like any further details or explanations? Here are five questions that might deepen your understanding:

1. How do we determine which function is the outer and which is the inner in a composition?
2. Can you reverse the process and find the composition of two functions to recreate the original function?
3. Why is it important that both functions are nontrivial?
4. What happens if you switch \( f(x) \) and \( g(x) \) in the composition?
5. How would the functions change if we used a different root, say a cube root, in the original expression?

### Tip:
Always start by identifying the simplest function in a composition problem—this is often the outer function!

Find two nontrivial functions f(x) and g(x) so that f(g(x)) = sqrt[5]((x + 1) / (x + 9)).

Learn how to find two nontrivial functions f(x) and g(x) where f(g(x)) results in a fifth root of a rational expression. This step-by-step guide covers function composition and radicals, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation