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### Problem Statement
We are given two functions:
- \( f(x) = x^2 \)
- \( g(x) = \sqrt{16 - x} - 2 \)

We are asked to find the composite function \( \left(\frac{f}{g}\right)(x) \) and state its domain in interval notation.

### Solution

#### Step 1: Determine the Composite Function \( \left(\frac{f}{g}\right)(x) \)
The composite function \( \left(\frac{f}{g}\right)(x) \) is defined as:
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
\]
Substituting the given functions \( f(x) \) and \( g(x) \):
\[
\left(\frac{f}{g}\right)(x) = \frac{x^2}{\sqrt{16 - x} - 2}
\]

#### Step 2: Determine the Domain
The domain of the function \( \left(\frac{f}{g}\right)(x) \) is determined by the following conditions:
1. **The denominator \( g(x) \) must not be zero:**
   \[
   g(x) = \sqrt{16 - x} - 2 \neq 0
   \]
   Solving \( \sqrt{16 - x} = 2 \):
   \[
   \sqrt{16 - x} = 2 \implies 16 - x = 4 \implies x = 12
   \]
   Therefore, \( x \neq 12 \).

2. **The expression under the square root in \( g(x) \) must be non-negative:**
   \[
   16 - x \geq 0 \implies x \leq 16
   \]
   This gives \( x \) must be in the interval \( (-\infty, 16] \).

Combining these two conditions, the domain is:
\[
(-\infty, 12) \cup (12, 16]
\]

### Final Answer
The composite function \( \left(\frac{f}{g}\right)(x) = \frac{x^2}{\sqrt{16 - x} - 2} \) has the domain \( (-\infty, 12) \cup (12, 16] \).

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Would you like more details or have any questions about this solution? 

Here are five related questions to explore further:
1. How would the domain change if \( f(x) \) was \( x^3 \) instead of \( x^2 \)?
2. Can you find the domain of \( g(x) = \sqrt{16 - x} - 2 \) alone?
3. How does the behavior of \( \left(\frac{f}{g}\right)(x) \) change near \( x = 12 \)?
4. What would happen to the domain if the square root in \( g(x) \) was replaced with a cube root?
5. How do you determine the domain of a function involving logarithms?

**Tip:** Always check the conditions under which the denominator of a fraction and the expression under a square root are valid to ensure the correct domain is determined.

Let f(x) = x^2 and g(x) = √(16 - x) - 2. Find (f/g)(x) and state its domain in interval notation.

Learn how to find the composite function (f/g)(x) where f(x) = x^2 and g(x) = √(16 - x) - 2. This problem explores function operations, domain restrictions involving square roots, and rational functions, making it suitable for students in Grades 10-12.

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