First slide

Functions (XII)

Question

If $f (2 x + \frac{y}{8}, 2 x - \frac{y}{8}) = x y,$ then $f (m, n) + f (n, m) = 0$

Moderate

A

$only when m = n$
B

$only when m \neq n$
C

$only when m = - n$
D

$for all m and n$

Solution

$\begin{array}{l} Let 2 x + \frac{y}{8} = α and 2 x - \frac{y}{8} = β . \\ Then x = \frac{α + β}{4} and y = 4 (α - β) . \\ Given f (2 x + \frac{y}{8}, 2 x - \frac{y}{8}) = x y \\ ∴ f (α, β) = α^{2} - β^{2} \\ or f (m, n) + f (n, m) = m^{2} - n^{2} + n^{2} - m^{2} = 0 for all m, n \end{array}$

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