To find functions $f(x)$ and $g(x)$ such that $(f \circ g)(x) = H(x)$, where $H(x) = |4x + 6|$, we can follow these steps:

1. Start with $H(x)$: $H(x) = |4x + 6|$ We aim to express $H(x)$ as $(f \circ g)(x) = f(g(x))$, which means $H(x)$ should be represented as a composition of two functions $f$ and $g$.

2. Define $g(x)$: Let $g(x)$ represent the inner function. A natural choice for $g(x)$ is the expression inside the absolute value: $g(x) = 4x + 6$ So, now we have $g(x) = 4x + 6$.

3. Define $f(x)$: The outer function $f(x)$ should apply the absolute value to the result of $g(x)$. Thus, we define $f(x)$ as: $f(x) = |x|$ This ensures that when we apply $f$ to $g(x)$, we get $H(x)$.

4. Check the composition: Now, let's check if $(f \circ g)(x) = f(g(x))$ gives us $H(x)$: $f(g(x)) = f(4x + 6) = |4x + 6|$ which matches $H(x)$.

Thus, the functions are:

• $g(x) = 4x + 6$
• $f(x) = |x|$

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How would the composition change if $H(x)$ did not include an absolute value?
2. Can you generalize this approach to other types of transformations (e.g., square roots)?
3. What happens to the composition if we replace $4x + 6$ with a different linear function?
4. How does this approach work if $f(x)$ and $g(x)$ are non-linear functions?
5. How would the composition change if we considered $H(x) = -(4x + 6)$ instead?

Tip: When solving composition problems, it's useful to identify the "inner" and "outer" operations by analyzing the structure of the function.