let f(x)


Let $f(x)=a^{x}(a>0)$ be written as $f(x)=f_{1}(x)+f_{2}(x)$, where $f_{1}(x)$ is an even function and $f_{2}(x)$ is an odd function. Then $f_{1}(x+y)+f_{1}(x-y)$ equals :

  1. (1) $2 f_{1}(x) f_{1}(y)$

  2. (2) $2 f_{1}(x+y) f_{1}(x-y)$

  3. (3) $2 f_{1}(x) f_{2}(y)$

  4. (4) $2 f_{1}(x+y) f_{2}(x-y)$

Correct Option: 1


Given function can be written as


where $f_{1}(x)=\frac{a^{x}+a^{-x}}{2}$ is even function

$f_{2}(x)=\frac{a^{x}-a^{-x}}{2}$ is odd function

$\Rightarrow f_{1}(x+y)+f_{1}(x-y)$



$=\frac{\left(a^{x}+a^{-x}\right)\left(a^{y}+a^{-y}\right)}{2}=2 f_{1}(x) \cdot f_{1}(y)$

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