If `f(x)=|x-2|" and "g(x)=f(f(x)),` then for `2ltxlt4,g'(x)` equals

Options

A. -1
B. 1
C. 0
D. none of these

Explanation

To solve the problem step by step, we start with the given functions:

1. **Define the functions**:
- \( f(x) = |x - 2| \)
- \( g(x) = f(f(x)) \)

2. **Analyze \( f(x) \)**:
- The function \( f(x) = |x - 2| \) can be expressed piecewise:
- For \( x < 2 \), \( f(x) = 2 - x \)
- For \( x \geq 2 \), \( f(x) = x - 2 \)

3. **Determine \( g(x) \)**:
- We need to find \( g(x) = f(f(x)) \).
- For the interval \( 2 < x < 4 \), since \( x \) is greater than 2, we have:
- \( f(x) = x - 2 \)
- Now we need to evaluate \( f(f(x)) \):
- \( f(f(x)) = f(x - 2) \)
- Since \( 2 < x < 4 \) implies \( 0 < x - 2 < 2 \), we use the piecewise definition:
- \( f(x - 2) = 2 - (x - 2) = 4 - x \)

4. **So, we have**:
- \( g(x) = 4 - x \) for \( 2 < x < 4 \).

5. **Differentiate \( g(x) \)**:
- To find \( g'(x) \), we differentiate \( g(x) = 4 - x \):
- \( g'(x) = -1 \)

6. **Conclusion**:
- Therefore, for \( 2 < x < 4 \), \( g'(x) = -1 \).

### Final Answer:
For \( 2 < x < 4 \), \( g'(x) = -1 \).