If f(x + y, x-y) = xy then the arithmetic mean of f(x, y) and f(y, x) is

Options

A. x
B. y
C. 0
D. `(x^(2) - y^(2))`

Answer

To solve the problem, we need to find the arithmetic mean of \( f(x, y) \) and \( f(y, x) \) given that \( f(x + y, x - y) = xy \).

### Step-by-Step Solution:

1. **Understanding the Function**:
We start with the function defined as:
\[ f(x + y, x - y) = xy \]

2. **Substituting Variables**:
To find \( f(x, y) \), we can set:
- Let \( a = x + y \)
- Let \( b = x - y \)

Thus, we can express \( x \) and \( y \) in terms of \( a \) and \( b \):
\[ x = \frac{a + b}{2}, \quad y = \frac{a - b}{2} \]

3. **Finding \( f(x, y) \)**:
We need to express \( f(x, y) \) in terms of \( a \) and \( b \):
\[ f(x, y) = f\left(\frac{a + b}{2}, \frac{a - b}{2}\right) \]
Now substituting these into the original function:
\[ f\left(\frac{a + b}{2}, \frac{a - b}{2}\right) = \left(\frac{a + b}{2}\right)\left(\frac{a - b}{2}\right) = \frac{(a + b)(a - b)}{4} = \frac{a^2 - b^2}{4} \]

4. **Finding \( f(y, x) \)**:
Similarly, we can find \( f(y, x) \):
\[ f(y, x) = f\left(\frac{a - b}{2}, \frac{a + b}{2}\right) = \left(\frac{a - b}{2}\right)\left(\frac{a + b}{2}\right) = \frac{(a - b)(a + b)}{4} = \frac{a^2 - b^2}{4} \]

5. **Calculating the Arithmetic Mean**:
Now we can calculate the arithmetic mean of \( f(x, y) \) and \( f(y, x) \):
\[ \text{Arithmetic Mean} = \frac{f(x, y) + f(y, x)}{2} = \frac{\frac{a^2 - b^2}{4} + \frac{a^2 - b^2}{4}}{2} = \frac{2 \cdot \frac{a^2 - b^2}{4}}{2} = \frac{a^2 - b^2}{4} \]

6. **Final Result**:
Since \( a = x + y \) and \( b = x - y \), we can substitute back:
\[ a^2 - b^2 = (x+y)^2 - (x-y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 4xy \]
Thus, the arithmetic mean simplifies to:
\[ \frac{4xy}{4} = xy \]

### Conclusion:
The arithmetic mean of \( f(x, y) \) and \( f(y, x) \) is:
\[ \boxed{0} \]