First slide

Algebra of real valued functions

Question

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

Moderate

A

x
B

y
C

0
D

xy

Solution

Let x + y = p, x - y = q. Then
$f (p, q) = \frac{p + q}{2} \cdot \frac{p - q}{2} = \frac{p^{2} - q^{2}}{4}$
$∴ f (x, y) = \frac{x^{2} - y^{2}}{4} a n d f (y, x) = \frac{y^{2} - x^{2}}{4}$
A.M.= $\frac{f (x, y) + f (y, x)}{2}$ =0

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