Let's evaluate each expression step-by-step.

Given Function Definitions:

1. $f(x) = \sqrt{x} + 5$
2. $g(x) = 4x - 8$
3. $h(x) = 7x$

Now, let's evaluate each expression in sequence.

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Expression 1: $f(g(5))$

1. Evaluate $g(5)$: $g(5) = 4(5) - 8 = 20 - 8 = 12$

2. Substitute into $f(x)$: $f(g(5)) = f(12) = \sqrt{12} + 5$

   Simplify: $f(12) = \sqrt{12} + 5 \approx 3.464 + 5 = 8.464$

So, $f(g(5)) \approx 8.464$.

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Expression 2: $g(f(87))$

1. Evaluate $f(87)$: $f(87) = \sqrt{87} + 5$

   Approximate $\sqrt{87}$: $f(87) \approx 9.327 + 5 = 14.327$

2. Substitute into $g(x)$: $g(f(87)) = g(14.327) = 4(14.327) - 8$

   Multiply and simplify: $g(14.327) \approx 57.308 - 8 = 49.308$

So, $g(f(87)) \approx 49.308$.

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Expression 3: $h(g(f(2)))$

1. Evaluate $f(2)$: $f(2) = \sqrt{2} + 5$

   Approximate $\sqrt{2}$: $f(2) \approx 1.414 + 5 = 6.414$

2. Substitute into $g(x)$: $g(f(2)) = g(6.414) = 4(6.414) - 8$

   Multiply and simplify: $g(6.414) \approx 25.656 - 8 = 17.656$

3. Substitute into $h(x)$: $h(g(f(2))) = h(17.656) = 7(17.656)$

   Multiply: $h(17.656) \approx 123.592$

So, $h(g(f(2))) \approx 123.592$.

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Final Answers:

1. $f(g(5)) \approx 8.464$
2. $g(f(87)) \approx 49.308$
3. $h(g(f(2))) \approx 123.592$

Would you like further details on any step?

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Here are five related questions to extend your understanding:

1. What happens if we replace $f(x)$ with $f(x) = x^2 + 5$ and re-evaluate these expressions?
2. How would the answers change if we used $g(x) = x^3 - 8$ instead?
3. If $h(x) = 2x + 3$, what would be the result of $h(g(f(5)))$?
4. Can we find $f(g(x))$ as a function of $x$ instead of specific values?
5. How can these compositions of functions be visualized graphically?

Tip: When working with function compositions, breaking down each step carefully helps prevent errors and simplifies complex expressions.