We are given three functions:

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

We are tasked with finding $f(g(h(x)))$. This means we will substitute $h(x)$ into $g(x)$, then substitute that result into $f(x)$. Let's solve this step-by-step:

Step 1: Compute $h(x)$

$h(x) = \sqrt{x}$

Step 2: Compute $g(h(x))$

Substitute $h(x) = \sqrt{x}$ into $g(x)$: $g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 8$

Step 3: Compute $f(g(h(x)))$

Now, substitute $g(h(x)) = \sqrt{x} - 8$ into $f(x)$: $f(g(h(x))) = f(\sqrt{x} - 8) = (\sqrt{x} - 8)^4 + 8$

Thus, the expression simplifies to: $f(g(h(x))) = (\sqrt{x} - 8)^4 + 8$

This is the final expression for $f(g(h(x)))$.

Would you like to expand this further or need a detailed breakdown of how to expand $(\sqrt{x} - 8)^4$?

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Related Questions:

1. What happens if we evaluate $f(g(h(16)))$?
2. How would the function $f(g(x))$ change if $g(x) = x + 8$?
3. What is the domain of $h(x) = \sqrt{x}$?
4. How can $(\sqrt{x} - 8)^4$ be expanded?
5. How would we find $f(g(x))$ if $f(x)$ were a different function, such as $f(x) = x^3 + 8$?

Tip: When dealing with compositions of functions, always start from the innermost function and work your way outward step by step.