Let's evaluate each of the given expressions step-by-step using the provided function definitions:

• $f(x) = \sqrt{x} + 4$
• $g(x) = 3x - 8$
• $h(x) = 7x$

1. $f(g(4))$

First, we need to compute $g(4)$: $g(4) = 3(4) - 8 = 12 - 8 = 4$

Now, substitute $g(4) = 4$ into $f(x)$: $f(4) = \sqrt{4} + 4 = 2 + 4 = 6$

So, $f(g(4)) = 6$.

2. $g(f(85))$

First, compute $f(85)$: $f(85) = \sqrt{85} + 4 \approx 9.22 + 4 = 13.22$

Now, substitute $f(85) \approx 13.22$ into $g(x)$: $g(13.22) = 3(13.22) - 8 = 39.66 - 8 = 31.66$

So, $g(f(85)) \approx 31.66$.

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

First, compute $f(2)$: $f(2) = \sqrt{2} + 4 \approx 1.41 + 4 = 5.41$

Now, substitute $f(2) \approx 5.41$ into $g(x)$: $g(5.41) = 3(5.41) - 8 = 16.23 - 8 = 8.23$

Finally, substitute $g(f(2)) \approx 8.23$ into $h(x)$: $h(8.23) = 7(8.23) = 57.61$

So, $h(g(f(2))) \approx 57.61$.

Final Results:

• $f(g(4)) = 6$
• $g(f(85)) \approx 31.66$
• $h(g(f(2))) \approx 57.61$

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. What is the domain of the function $f(x)$?
2. How do function compositions affect the domain of the resulting function?
3. Can you compute $f(g(10))$ using the same method?
4. What happens if you reverse the composition, such as $g(h(x))$?
5. What are the practical applications of function compositions?

Tip: Always check if the input to a function lies within its domain before computing its value!