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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, \( h(g(f(2))) \approx 0.623 \).

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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.

Evaluate each of the following expressions given the function definitions for f(x), g(x), and h(x): a. f(g(3)), b. g(f(81)), c. h(g(f(2))).

Learn how to evaluate function compositions using the definitions of f(x) = sqrt(x) + 5, g(x) = 3x - 8, and h(x) = 7/x. Explore examples of f(g(3)), g(f(81)), and h(g(f(2))). This problem covers key concepts like square root functions, linear functions, and rational functions, suitable for students in Grades 9-12.

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