https://file.aiquickdraw.com/mathbot/file/17268457588976bf1tk6n.jpg

We are given three functions \( f(x) = x^4 + 3 \), \( g(x) = x - 6 \), and \( h(x) = \sqrt{x} \). We need to find \( (f \circ g \circ h)(x) \), which means we are looking for the composite function \( f(g(h(x))) \).

Let's break it down step by step:

1. First, apply \( h(x) \):
   \[
   h(x) = \sqrt{x}
   \]
   
2. Next, apply \( g(x) \) to \( h(x) \):
   \[
   g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 6
   \]

3. Finally, apply \( f(x) \) to \( g(h(x)) \):
   \[
   f(g(h(x))) = f(\sqrt{x} - 6)
   \]
   Now substitute \( \sqrt{x} - 6 \) into \( f(x) = x^4 + 3 \):
   \[
   f(\sqrt{x} - 6) = (\sqrt{x} - 6)^4 + 3
   \]

Thus, the composite function \( (f \circ g \circ h)(x) \) is:
\[
(f \circ g \circ h)(x) = (\sqrt{x} - 6)^4 + 3
\]

Would you like further details on this process, or any questions related to the steps?

Here are five related questions you might find interesting:
1. What are the steps to find the domain of \( f(g(h(x))) \)?
2. How would the expression change if \( h(x) \) were \( x^2 \) instead of \( \sqrt{x} \)?
3. How would you expand \( (\sqrt{x} - 6)^4 \) using binomial expansion?
4. Can \( f(g(h(x))) \) be simplified further in some special cases?
5. What happens to \( f(g(h(x))) \) as \( x \to \infty \)?

**Tip**: Always evaluate the domain of composite functions to ensure no domain violations occur during substitutions!

If f(x) = x^4 + 3, g(x) = x - 6, h(x) = √x, then f(g(h(x))) = ?

Learn how to compute the composite function f(g(h(x))) where f(x) = x^4 + 3, g(x) = x - 6, and h(x) = √x. This step-by-step solution explores function composition, polynomials, and radicals, perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation