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To solve \( f \circ g \circ h(x) \), we need to evaluate the composition of the functions in the order they appear, starting with the innermost function and proceeding outward. Let's go through the steps:

1. First, calculate \( h(x) \):
   \[
   h(x) = \sqrt{x}
   \]

2. Next, find \( g(h(x)) \):
   \[
   g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 7
   \]

3. Finally, calculate \( f(g(h(x))) \):
   \[
   f(g(h(x))) = f(\sqrt{x} - 7) = (\sqrt{x} - 7)^4 + 5
   \]

Thus, \( f \circ g \circ h(x) = (\sqrt{x} - 7)^4 + 5 \).

Would you like to explore more about function compositions or have any other questions about this problem? Here are some related topics to consider:

1. What happens if you change the order of composition?
2. How does the graph of \( f \circ g \circ h(x) \) compare to the individual graphs of \( f(x) \), \( g(x) \), and \( h(x) \)?
3. Can you derive a general method for simplifying nested function compositions?
4. What are the domain considerations for the composition \( f \circ g \circ h(x) \)?
5. How does modifying the functions \( f \), \( g \), or \( h \) affect the final output?

Tip: When composing functions, always consider the domain restrictions imposed by each function, as this can impact the valid input values for the composition.

If f(x) = x^4 + 5, g(x) = x - 7, h(x) = √x, then f ∘ g ∘ h(x) = ?

This solution guides you through solving a composite function involving f(x) = x^4 + 5, g(x) = x - 7, and h(x) = √x. Learn how to work step-by-step with function compositions, exponents, and square roots, suitable for high school students.

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