Question

If $f(x)=\frac{a^x+a^{-x}}{2}$ and $f(x+y)+f(x-y)=K f(x)$ $f(y)$ then $K$ is equal to

• A. 2
• B. 4
• C. -2
• D. none of these

Answer 1

Answer: (A)

$f(x+y)+f(x-y) $
$=\frac{a^{x+y}+a^{-(x+y)}}{2}+\frac{a^{x-y}+a^{-(x-y)}}{2}$
$=\frac{a^x\left(a^y+a^{-y}\right)+a^{-x}\left(a^{-y}+a^y\right)}{2}$
$=\frac{\left(a^x+a^{-x}\right)\left(a^y+a^{-y}\right)}{2} $
$=2 \cdot f(x) f(y) \Rightarrow k=2 .$