We are given the functions $f(x) = 5x^4 - 8$, $g(x) = 5x - 9$, and $h(x) = \sqrt{x}$, and asked to find $(f \circ g \circ h)(x)$, which is the composition of the three functions.

The composition $(f \circ g \circ h)(x)$ can be written as: $(f \circ g \circ h)(x) = f(g(h(x)))$

Step-by-step process:

1. Start with $h(x)$: $h(x) = \sqrt{x}$

2. Next, substitute $h(x)$ into $g(x)$: $g(h(x)) = g(\sqrt{x}) = 5\sqrt{x} - 9$

3. Finally, substitute $g(h(x))$ into $f(x)$: $f(g(h(x))) = f(5\sqrt{x} - 9)$ Now, substitute $5\sqrt{x} - 9$ into $f(x) = 5x^4 - 8$: $f(5\sqrt{x} - 9) = 5(5\sqrt{x} - 9)^4 - 8$

This expression represents $(f \circ g \circ h)(x)$. However, simplifying $(5\sqrt{x} - 9)^4$ would require expansion, which can be done if necessary, but the current form represents the final composition.

Would you like me to simplify this further or clarify any part of the process?

Related Questions:

1. How do you simplify expressions involving nested functions?
2. What happens to the domain of a function when composing multiple functions?
3. Can you graph the result of the composite function?
4. How do you expand binomials raised to a power, such as $(5\sqrt{x} - 9)^4$?
5. What are some common errors to avoid when composing functions?

Tip:

When composing functions, always start from the innermost function and work outward, step by step.