Let's go through each function and the compositions of those functions based on the provided definitions:

1. Given functions:

   • $f(x) = \sqrt{x} + 5$
   • $g(x) = 3x - 6$
   • $h(x) = 5x$

2. Composition 1: $f(g(x))$

   This composition means you substitute $g(x)$ into $f(x)$.

   • Since $f(x) = \sqrt{x} + 5$, and $g(x) = 3x - 6$, substitute $g(x)$ into $f(x)$: $f(g(x)) = \sqrt{g(x)} + 5 = \sqrt{3x - 6} + 5$ This is correct.

3. Composition 2: $g(f(x))$

   This means you substitute $f(x)$ into $g(x)$.

   • Since $g(x) = 3x - 6$, and $f(x) = \sqrt{x} + 5$, substitute $f(x)$ into $g(x)$: $g(f(x)) = 3f(x) - 6 = 3(\sqrt{x} + 5) - 6 = 3\sqrt{x} + 15 - 6 = 3\sqrt{x} + 9$ This is the correct expression.

4. Composition 3: $h(f(x))$

   This means you substitute $f(x)$ into $h(x)$.

   • Since $h(x) = 5x$, and $f(x) = \sqrt{x} + 5$, substitute $f(x)$ into $h(x)$: $h(f(x)) = 5f(x) = 5(\sqrt{x} + 5) = 5\sqrt{x} + 25$ So, $h(f(x)) = 5\sqrt{x} + 25$.

Summary of the function compositions:

1. $f(g(x)) = \sqrt{3x - 6} + 5$
2. $g(f(x)) = 3\sqrt{x} + 9$
3. $h(f(x)) = 5\sqrt{x} + 25$

Would you like more details or have any questions about the process?

Here are 5 questions that expand on the topic:

1. What is the domain of $f(g(x))$?
2. How would you find $g(h(x))$?
3. Can you compute $f(f(x))$ using the given function?
4. What would be the inverse of $f(x)$?
5. How do you graph $h(f(x))$?

Tip: When dealing with function compositions, always carefully substitute the entire inner function into the outer function for clarity and accuracy.