Let `f(x)=|x-2|`, then

Options

A. `f(x^(2))=(f(x))^(2)`
B. `f(x+y)=f(x)f(y)`
C. `f(|x|)=|f(x)|`
D. none of these

Answer

To solve the problem, we need to analyze the function \( f(x) = |x - 2| \) and check the validity of the following options:

1. \( f(x^2) \) vs \( f(x)^2 \)
2. \( f(x+y) \) vs \( f(x) \cdot f(y) \)
3. \( f(|x|) \) vs \( |f(x)| \)

Let's evaluate each option step by step.

### Step 1: Evaluate \( f(x^2) \) and \( f(x)^2 \)

1. **Calculate \( f(x^2) \)**:
\[ f(x^2) = |x^2 - 2| \]

2. **Calculate \( f(x)^2 \)**:
\[ f(x) = |x - 2| \implies f(x)^2 = (|x - 2|)^2 = (x - 2)^2 \]

3. **Compare \( f(x^2) \) and \( f(x)^2 \)**:
- \( f(x^2) = |x^2 - 2| \)
- \( f(x)^2 = (x - 2)^2 \)

These two expressions are not equal for all \( x \). For example:
- If \( x = 0 \):
\[ f(0^2) = |0 - 2| = 2 \quad \text{and} \quad f(0)^2 = (|0 - 2|)^2 = 4 \]
- If \( x = 3 \):
\[ f(3^2) = |9 - 2| = 7 \quad \text{and} \quad f(3)^2 = (|3 - 2|)^2 = 1 \]

Thus, **Option 1 is incorrect**.

### Step 2: Evaluate \( f(x+y) \) and \( f(x) \cdot f(y) \)

1. **Calculate \( f(x+y) \)**:
\[ f(x+y) = |(x+y) - 2| = |x + y - 2| \]

2. **Calculate \( f(x) \cdot f(y) \)**:
\[ f(x) = |x - 2| \quad \text{and} \quad f(y) = |y - 2| \implies f(x) \cdot f(y) = |x - 2| \cdot |y - 2| \]

3. **Compare \( f(x+y) \) and \( f(x) \cdot f(y) \)**:
- For \( x = 0 \) and \( y = 0 \):
\[ f(0 + 0) = |0 - 2| = 2 \quad \text{and} \quad f(0) \cdot f(0) = 2 \cdot 2 = 4 \]
- For \( x = 2 \) and \( y = 2 \):
\[ f(2 + 2) = |4 - 2| = 2 \quad \text{and} \quad f(2) \cdot f(2) = 0 \cdot 0 = 0 \]

Thus, **Option 2 is incorrect**.

### Step 3: Evaluate \( f(|x|) \) and \( |f(x)| \)

1. **Calculate \( f(|x|) \)**:
\[ f(|x|) = ||x| - 2| \]

2. **Calculate \( |f(x)| \)**:
\[ |f(x)| = ||x - 2| | = |x - 2| \quad \text{(since absolute value is always non-negative)} \]

3. **Compare \( f(|x|) \) and \( |f(x)| \)**:
- For \( x = 3 \):
\[ f(|3|) = |3 - 2| = 1 \quad \text{and} \quad |f(3)| = |1| = 1 \]
- For \( x = -3 \):
\[ f(|-3|) = |3 - 2| = 1 \quad \text{and} \quad |f(-3)| = |(-3 - 2)| = 5 \]

Thus, **Option 3 is incorrect**.

### Conclusion
Since all options are incorrect, the final answer is:
**None of these options are correct.**