https://file.aiquickdraw.com/mathbot/file/1726077835095bcyxfu6j.jpg

We are given that \( h(x) = (f \circ g)(x) = f(g(x)) \), where \( h(x) = \frac{3}{(x - 8)^7} \). We need to find functions \( f(x) \) and \( g(x) \) such that \( h(x) = f(g(x)) \).

### Step-by-step approach:

1. **Examine \( h(x) \)**:
   \[
   h(x) = \frac{3}{(x - 8)^7}
   \]
   It looks like the expression inside the denominator can be split into two functions. The \( (x - 8) \) part suggests one function, while raising to the power of 7 and dividing by 3 suggests another.

2. **Identify \( g(x) \)**:
   Since \( h(x) \) involves \( (x - 8) \), we can try setting \( g(x) = x - 8 \). This would simplify \( h(x) \) to a form that depends on \( f(x) \).

3. **Identify \( f(x) \)**:
   Now, substitute \( g(x) = x - 8 \) into \( h(x) \):
   \[
   h(x) = \frac{3}{(g(x))^7}
   \]
   Therefore, we can set \( f(x) = \frac{3}{x^7} \).

So, we have:
- \( f(x) = \frac{3}{x^7} \)
- \( g(x) = x - 8 \)

### Now, let's match this with the options:

- **A.** \( f(x) = \frac{3}{7x} \) and \( g(x) = x - 8 \): Incorrect, since the form of \( f(x) \) doesn't match.
- **B.** \( f(x) = \frac{7}{x} \) and \( g(x) = x - 8 \): Incorrect, as the power and constant don't match.
- **C.** \( f(x) = \frac{3}{x} \) and \( g(x) = (x - 8)^7 \): Incorrect, since \( g(x) \) is not raised to the seventh power before applying \( f \).
- **D.** \( f(x) = \frac{7}{x^3} \) and \( g(x) = (x - 8)^7 \): Incorrect as the powers and constants are not correct.

### Correct Answer:
None of the provided options match the correct solution, which is \( f(x) = \frac{3}{x^7} \) and \( g(x) = x - 8 \).

Do you want further explanation or have any other questions?

### Related questions:
1. How does function composition work in general?
2. How can you identify potential \( f(x) \) and \( g(x) \) in composite functions?
3. Why is \( g(x) = x - 8 \) the correct choice here?
4. How does the order of operations impact function composition?
5. What are other ways to decompose complex functions?

### Tip:
When working with function composition, try simplifying the outer function first and look for common patterns like shifts or powers to identify the inner function.

Find f(x) and g(x) such that h(x) = (f ∘ g)(x). h(x) = 3 / (x - 8)^7. Select all that apply.

Learn how to find functions f(x) and g(x) such that h(x) = f(g(x)) for the expression h(x) = 3 / (x - 8)^7. This problem involves function composition, algebra, and exponents. Suitable for high school 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