https://file.aiquickdraw.com/mathbot/file/1725993276560hdlemfkc.jpg

The problem given in the image is as follows:

You are asked to find two functions \( f(x) \) and \( g(x) \) such that the composition \( (f \circ g)(x) = H(x) \), where:

\[
H(x) = (7x - 6)^6
\]

Additionally, the identity function cannot be one of the answers, and there may be more than one correct solution.

### Strategy to Solve:

We are given \( H(x) = (7x - 6)^6 \), and we need to decompose this function into two functions \( f(x) \) and \( g(x) \) such that:

\[
(f \circ g)(x) = f(g(x)) = H(x) = (7x - 6)^6
\]

A common approach is to let \( g(x) \) handle the inner transformation of \( x \), while \( f(x) \) applies an outer transformation.

### Step-by-Step Solution:

1. **Choice for \( g(x) \):**
   - Let \( g(x) = 7x - 6 \). This simplifies the expression inside the parentheses of \( H(x) \).

2. **Choice for \( f(x) \):**
   - Since \( g(x) = 7x - 6 \), applying this to \( f(x) \) means we need \( f(g(x)) = (7x - 6)^6 \). Thus, we can choose \( f(x) = x^6 \) to raise the result of \( g(x) \) to the sixth power.

Now, let's verify:

\[
f(g(x)) = f(7x - 6) = (7x - 6)^6 = H(x)
\]

Thus, one possible pair of functions is:

\[
f(x) = x^6 \quad \text{and} \quad g(x) = 7x - 6
\]

### Final Answer:

- \( f(x) = x^6 \)
- \( g(x) = 7x - 6 \)

These two functions satisfy the composition condition \( (f \circ g)(x) = H(x) \).

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Would you like further explanation, or do you have any questions about this solution?

Here are 5 related questions for practice:

1. How would the functions \( f(x) \) and \( g(x) \) change if \( H(x) = (5x + 3)^4 \)?
2. Can you decompose \( H(x) = (2x + 1)^3 \) into functions \( f(x) \) and \( g(x) \)?
3. What are the benefits of function composition in solving real-world problems?
4. How can we check if two functions are inverses of each other using composition?
5. What happens if we reverse the order of composition, i.e., \( g(f(x)) \)?

**Tip:** Always start by identifying the inner and outer functions when solving function composition problems.

Suppose H(x) = (7x - 6)^6. Find two functions f and g such that (f ∘ g)(x) = H(x). Neither function can be the identity function.

Explore how to find two functions f(x) and g(x) such that their composition equals H(x) = (7x - 6)^6. This problem delves into function composition and algebraic manipulation, making it suitable for students in Grades 10-12.

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