The problem in the image asks you to find $(f \circ g \circ h)(x)$, given the following functions:

$f(x) = x^4 + 8, \quad g(x) = x - 7, \quad h(x) = \sqrt{x}.$

To find $(f \circ g \circ h)(x)$, we need to perform the composition of functions. This means that we evaluate $h(x)$ first, then $g(h(x))$, and finally $f(g(h(x)))$.

Step-by-Step Solution:

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

2. Next, apply $g(x)$ to $h(x)$: $g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 7.$

3. Finally, apply $f(x)$ to $g(h(x))$: $f(g(h(x))) = f(\sqrt{x} - 7) = (\sqrt{x} - 7)^4 + 8.$

   To expand $(\sqrt{x} - 7)^4$, you can apply the binomial theorem or expand step-by-step, but for now, you can leave the answer in this form: $f(g(h(x))) = (\sqrt{x} - 7)^4 + 8.$

Thus, the final answer for $(f \circ g \circ h)(x)$ is:

$(f \circ g \circ h)(x) = (\sqrt{x} - 7)^4 + 8.$

Would you like me to expand $(\sqrt{x} - 7)^4$ further? Let me know if you have any questions.

Here are 5 related questions you could explore:

1. What happens if we change $f(x)$ to a different function, like $f(x) = x^2 + 5$?
2. How do we compute $(g \circ f \circ h)(x)$ instead of $(f \circ g \circ h)(x)$?
3. What is the domain of $(f \circ g \circ h)(x)$?
4. Can we simplify $(f \circ g \circ h)(x)$ for specific values of $x$?
5. How do compositions of functions behave under different types of functions (e.g., linear, quadratic)?

Tip: When composing functions, always start from the innermost function and work your way out step-by-step.