https://file.aiquickdraw.com/mathbot/file/17285309764426izk7s8t.jpg

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.

Find (f ∘ g ∘ h)(x) given that f(x) = x^4 + 8, g(x) = x - 7, and h(x) = sqrt(x).

Learn how to find (f ∘ g ∘ h)(x) with detailed steps. The problem involves composing functions f(x) = x^4 + 8, g(x) = x - 7, and h(x) = sqrt(x), making use of radicals and polynomials. Suitable for college algebra students.

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