For which of the following functions does `f(x) f(2 - x)` ?

Options

A. `f(x) = x + 2`
B. `f(x) = 2x - x^2`
C. `f(x) = 2 - x`
D. `f(x) = (2 - x)^(2)`

Answer

To determine for which of the given functions \( f(x) \) satisfies the condition \( f(x) = f(2 - x) \), we will evaluate each function one by one.

### Step 1: Evaluate \( f(x) = x + 2 \)

1. **Calculate \( f(2 - x) \)**:
\[ f(2 - x) = (2 - x) + 2 = 4 - x \]

2. **Check if \( f(x) = f(2 - x) \)**:
\[ x + 2 \neq 4 - x \]
This is not true for all \( x \). Thus, this function does not satisfy the condition.

### Step 2: Evaluate \( f(x) = 2x - x^2 \)

1. **Calculate \( f(2 - x) \)**:
\[ f(2 - x) = 2(2 - x) - (2 - x)^2 \]
Expanding this:
\[ = 4 - 2x - (4 - 4x + x^2) = 4 - 2x - 4 + 4x - x^2 = 2x - x^2 \]

2. **Check if \( f(x) = f(2 - x) \)**:
\[ 2x - x^2 = 2x - x^2 \]
This is true for all \( x \). Thus, this function satisfies the condition.

### Step 3: Evaluate \( f(x) = 2 - x \)

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

2. **Check if \( f(x) = f(2 - x) \)**:
\[ 2 - x \neq x \]
This is not true for all \( x \). Thus, this function does not satisfy the condition.

### Step 4: Evaluate \( f(x) = (2 - x)^2 \)

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

2. **Check if \( f(x) = f(2 - x) \)**:
\[ (2 - x)^2 \neq x^2 \]
This is not true for all \( x \). Thus, this function does not satisfy the condition.

### Conclusion

The only function that satisfies \( f(x) = f(2 - x) \) is:
\[ \boxed{f(x) = 2x - x^2} \]