Question

If \( f\left(x + \frac{y}{8}, y - \frac{x}{8}\right) = xy \), then \( f(m,n) + f(n,m) = 0 \) is true:

• A. only when \( m = n \)
• B. only when \( m \neq n \)
• C. only when \( m = -n \)
• D. for all \( m \) and \( n \)

Hint:

When a function is symmetric in variables, test it by swapping them — this often reveals identities valid for all inputs.

Answer 1

The Correct Option is D

We are given: \[ f\left(x + \frac{y}{8}, y - \frac{x}{8}\right) = xy \] If we set \(x = m\) and \(y = n\): \[ f\left(m + \frac{n}{8}, n - \frac{m}{8}\right) = mn \] Now, interchange \(m\) and \(n\): \[ f\left(n + \frac{m}{8}, m - \frac{n}{8}\right) = nm = mn \] Adding the two expressions for \(f(m,n)\) and \(f(n,m)\) from these, we see that the sum will cancel to zero for all \(m, n\). Thus the statement is valid for \({\text{all } m, n}\).