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We are asked to evaluate \( f(g(h(x))) \) given the following functions:

\[
f(x) = x^4 + 6, \quad g(x) = x - 5, \quad h(x) = \sqrt{x}.
\]

Let's break this down step by step.

### Step 1: Evaluate \( h(x) \)
\[
h(x) = \sqrt{x}.
\]

### Step 2: Apply \( g(x) \) to \( h(x) \)
We need to substitute \( h(x) = \sqrt{x} \) into \( g(x) = x - 5 \).
\[
g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 5.
\]

### Step 3: Apply \( f(x) \) to \( g(h(x)) \)
Now substitute \( g(h(x)) = \sqrt{x} - 5 \) into \( f(x) = x^4 + 6 \).
\[
f(g(h(x))) = f(\sqrt{x} - 5) = (\sqrt{x} - 5)^4 + 6.
\]

Thus, the final expression for \( f(g(h(x))) \) is:
\[
f(g(h(x))) = (\sqrt{x} - 5)^4 + 6.
\]

Would you like further details on how to expand or simplify this result? Here are a few related questions you can consider:

1. What is the expanded form of \( (\sqrt{x} - 5)^4 \)?
2. How do you compose functions in general?
3. What are some applications of composite functions in real-life problems?
4. How can you verify if two functions are inverses using compositions?
5. What happens if we change one of the functions in the composition? 

**Tip:** When working with nested functions, start from the innermost function and work your way outward to avoid mistakes.

If f(x) = x^4 + 6, g(x) = x - 5, and h(x) = √x, then find f(g(h(x)))?

Learn how to solve the composite function f(g(h(x))) using the functions f(x) = x^4 + 6, g(x) = x - 5, and h(x) = √x. Follow a step-by-step breakdown of each function's application, from evaluating h(x) to simplifying f(g(h(x))). Suitable for high school students in Grades 10-12.

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