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To solve \( (f \circ f)(x) \), which represents the composite function \( f(f(x)) \), we will substitute the definition of \( f(x) \) into itself.

### Given:
- \( f(x) = \sqrt{x - 2} \)

### Step-by-Step Solution:
1. Start with \( f(f(x)) \). This means substituting \( f(x) = \sqrt{x - 2} \) into the function \( f \):
   \[
   f(f(x)) = f\left(\sqrt{x - 2}\right)
   \]
2. Now, apply the definition of \( f(x) \) again. Substitute \( \sqrt{x - 2} \) into the function \( f \), which is \( f(x) = \sqrt{x - 2} \). This gives:
   \[
   f(f(x)) = \sqrt{\sqrt{x - 2} - 2}
   \]
3. Simplify the expression inside the second square root:
   \[
   f(f(x)) = \sqrt{\sqrt{x - 2} - 2}
   \]
   There is no further simplification possible for this expression.

### Final Answer:
\[
f(f(x)) = \sqrt{\sqrt{x - 2} - 2}
\]

Let me know if you'd like more details or further clarifications.

Here are five related questions you might find interesting:
1. How do you find the composite function of \( g \circ f \) given \( g(x) = 3x^2 + 2 \)?
2. What are the domain restrictions for the function \( f(f(x)) \)?
3. How do you solve for \( f(g(x)) \) given the functions in this problem?
4. How can you apply this concept of composite functions in real-life applications?
5. What happens if you compute \( (h \circ f)(x) \) given \( h(x) = \frac{3}{x + 2} \)?

**Tip:** When dealing with composite functions, always pay close attention to the domain restrictions of both functions to ensure valid inputs at each step.

Find the value of the composite function: (f ∘ f)(x) given f(x) = √(x - 2), g(x) = 3x² + 2, and h(x) = 3/(x + 2).

Learn how to solve the composite function (f ∘ f)(x) for f(x) = √(x - 2). This step-by-step guide covers the key mathematical concepts and formulas for finding the composite function. Suitable for students in Grades 10-12.

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