Let's break down the problem step by step:

Given Functions:

1. $f(x) = x^4 + 3$
2. $g(x) = x - 2$
3. $h(x) = \sqrt{x}$

You are asked to find $f(g(h(x)))$.

Steps:

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

2. Substitute $h(x)$ into $g(x)$: $g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 2$

3. Substitute $g(h(x))$ into $f(x)$: $f(g(h(x))) = f(\sqrt{x} - 2)$ Now substitute $\sqrt{x} - 2$ into $f(x) = x^4 + 3$: $f(\sqrt{x} - 2) = (\sqrt{x} - 2)^4 + 3$

4. Simplify $(\sqrt{x} - 2)^4$: This expression is more complicated and expands into multiple terms, but for now, the most simplified form is: $f(g(h(x))) = (\sqrt{x} - 2)^4 + 3$ You can further expand $(\sqrt{x} - 2)^4$ if required, but the general form is above.

Would you like more details on the expansion, or do you have any other questions?

Here are 5 related questions for deeper understanding:

1. How do you expand $(\sqrt{x} - 2)^4$ step by step?
2. What happens if one of the functions $f(x), g(x), h(x)$ is a different type of polynomial?
3. How do you compute the composition of functions when dealing with trigonometric functions?
4. What is the effect of domain restrictions when composing functions involving square roots?
5. How can we verify if a composite function is invertible?

Tip: When solving composition of functions, work from the innermost function outward.