Let's break down and solve each of the expressions based on the given functions:

The functions are:

• $f(x) = \sqrt{x} + 5$
• $g(x) = 3x - 8$
• $h(x) = \frac{7}{x}$

a. $f(g(3))$

1. First, find $g(3)$: $g(3) = 3(3) - 8 = 9 - 8 = 1$
2. Now, substitute $g(3) = 1$ into $f(x)$: $f(1) = \sqrt{1} + 5 = 1 + 5 = 6$ So, $f(g(3)) = 6$.

b. $g(f(81))$

1. First, find $f(81)$: $f(81) = \sqrt{81} + 5 = 9 + 5 = 14$
2. Now, substitute $f(81) = 14$ into $g(x)$: $g(14) = 3(14) - 8 = 42 - 8 = 34$ So, $g(f(81)) = 34$.

c. $h(g(f(2)))$

1. First, find $f(2)$: $f(2) = \sqrt{2} + 5 \approx 1.414 + 5 = 6.414$
2. Now, substitute $f(2)$ into $g(x)$: $g(6.414) = 3(6.414) - 8 = 19.242 - 8 = 11.242$
3. Finally, substitute $g(f(2))$ into $h(x)$: $h(11.242) = \frac{7}{11.242} \approx 0.623$ So, $h(g(f(2))) \approx 0.623$.

Would you like further details or have any other questions?

Here are 5 related questions to explore further:

1. How would the results change if $g(x)$ was a different linear function?
2. What is the domain of the composite function $f(g(x))$?
3. Can you find the inverse of any of the given functions?
4. How does $h(g(f(x)))$ behave for negative values of $x$?
5. What would $f(g(x))$ look like graphically?

Tip: Always check the domain of each function before performing operations, especially when square roots or division by zero are involved.