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The image provided shows the problem asking to find the formula for the new function formed by the composition \( (f \circ g \circ h)(x) \), where the given functions are:

\[
f(x) = \sqrt{x+3}, \quad g(x) = \frac{6}{x}, \quad h(x) = x^2.
\]

Let's break down the composition step-by-step:

### 1. Compute \( (g \circ h)(x) \):
We need to substitute \( h(x) = x^2 \) into \( g(x) = \frac{6}{x} \):

\[
g(h(x)) = g(x^2) = \frac{6}{x^2}.
\]

### 2. Now, compute \( (f \circ g \circ h)(x) \):
Substitute \( g(h(x)) = \frac{6}{x^2} \) into \( f(x) = \sqrt{x + 3} \):

\[
f(g(h(x))) = f\left(\frac{6}{x^2}\right) = \sqrt{\frac{6}{x^2} + 3}.
\]

Thus, the composed function is:

\[
(f \circ g \circ h)(x) = \sqrt{\frac{6}{x^2} + 3}.
\]

### Domain:
To determine the domain, we need to consider the restrictions of each function:

- \( h(x) = x^2 \) is defined for all real \( x \), \( (-\infty, \infty) \).
- \( g(x) = \frac{6}{x} \) has a restriction that \( x \neq 0 \).
- \( f(x) = \sqrt{x+3} \) requires that \( x + 3 \geq 0 \), or \( x \geq -3 \).

Thus, after combining these conditions, the domain will be restricted by \( h(x) \neq 0 \), meaning \( x \neq 0 \). However, since \( h(x) = x^2 \), \( h(x) \) is always non-negative, and the square root function restricts it further.

Therefore, the domain is:

\[
(0, \infty).
\]

Let me know if you would like further clarifications or detailed explanations!

### Follow-up questions:
1. How is the domain determined in function compositions?
2. What are the possible cases where the domain of a composed function could be restricted further?
3. How do you handle square roots in function composition?
4. Can you explain how the square in \( h(x) = x^2 \) affects the domain of the composition?
5. Why does \( g(x) = \frac{6}{x} \) cause issues when \( x = 0 \)?

### Tip:
Always evaluate the inner functions first in function composition and ensure the domains align before simplifying the entire expression.

Given the functions f(x) = √(x + 3), g(x) = 6/x, and h(x) = x^2, write a simplified formula for the new function based on (f ∘ g ∘ h)(x) and determine its domain.

Learn how to simplify the composition of functions f(x) = √(x + 3), g(x) = 6/x, and h(x) = x^2. This detailed solution explains each step and determines the domain of the composed function, covering key concepts such as square roots, rational functions, and domain restrictions. Suitable for high school students in Grades 10-12.

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