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 evaluate the four given options one by one.

### Step 1: Evaluate Option 1
**Option 1:** \( f(x^2) = f(x)^2 \)

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

2. Calculate \( f(x)^2 \):
\[ f(x) = |x - 2| \quad \Rightarrow \quad f(x)^2 = (|x - 2|)^2 = (x - 2)^2 \]

3. Compare \( f(x^2) \) and \( f(x)^2 \):
- \( |x^2 - 2| \) is not equal to \( (x - 2)^2 \) for all \( x \).
- For example, if \( x = 0 \):
\[ f(0^2) = |0 - 2| = 2 \quad \text{and} \quad f(0)^2 = |0 - 2|^2 = 4 \]
- Therefore, \( f(x^2) \neq f(x)^2 \).

**Conclusion for Option 1:** Incorrect.

### Step 2: Evaluate Option 2
**Option 2:** \( f(x + y) = f(x) \cdot f(y) \)

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

2. Calculate \( f(x) \cdot f(y) \):
\[ f(x) \cdot f(y) = |x - 2| \cdot |y - 2| \]

3. Compare \( f(x + y) \) and \( f(x) \cdot f(y) \):
- These two expressions are not equal for all \( x \) and \( y \).
- For example, if \( x = 0 \) and \( y = 0 \):
\[ f(0 + 0) = |0 - 2| = 2 \quad \text{and} \quad f(0) \cdot f(0) = 2 \cdot 2 = 4 \]
- Therefore, \( f(x + y) \neq f(x) \cdot f(y) \).

**Conclusion for Option 2:** Incorrect.

### Step 3: Evaluate Option 3
**Option 3:** \( f(|x|) = |f(x)| \)

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

2. Calculate \( |f(x)| \):
\[ f(x) = |x - 2| \quad \Rightarrow \quad |f(x)| = ||x - 2|| \]
- Since \( |x - 2| \) is always non-negative, we have \( |f(x)| = |x - 2| \).

3. Compare \( f(|x|) \) and \( |f(x)| \):
- If \( x \geq 0 \):
\[ f(|x|) = ||x| - 2| = |x - 2| \quad \text{and} \quad |f(x)| = |x - 2| \]
- If \( x < 0 \):
\[ f(|x|) = ||x| - 2| = |-x - 2| = |x + 2| \quad \text{and} \quad |f(x)| = |x - 2| \]
- Both cases show that \( f(|x|) = |f(x)| \).

**Conclusion for Option 3:** Correct.

### Step 4: Evaluate Option 4
**Option 4:** None of these.

Since we have found that Option 3 is correct, we can conclude that Option 4 is not applicable.

### Final Conclusion
- **Option 1:** Incorrect
- **Option 2:** Incorrect
- **Option 3:** Correct
- **Option 4:** Incorrect