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 $(f \circ g)(x) = H(x)$.

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.