Question

If the function $f\left(x\right)=\frac{a^x+a^{-x}}{2},(a>2)$, then $f(x+y)+f(x-y)$ is equal to

• A. $2f\left(x\right)f\left(y\right)$
• B. $f\left(x\right)f\left(y\right)$
• C. $\frac{f\left(x\right)}{f\left(y\right)}$
• D. None of these

Answer 1

The correct option is A $2f\left(x\right)f\left(y\right)$

$f\left(x\right)=\frac{a^x+a^{-x}}{2}$

$f(x+y)=\frac{a^{x+y}+a^{-(x+y)}}{2},f(x-y)=\frac{a^{x-y}+a^{-(x-y)}}{2}$

$f(x+y)+f(x-y)=\frac{1}{2}(a^x{a}^y+a^{-x}{a}^{-y}+a^x{a}^{-y}+a^{-x}{a}^y)$

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

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

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

$=\frac{1}{2}2f\left(x\right)2f\left(y\right)$

$=2f\left(x\right).f\left(y\right)$

$\therefore f(x+y)+f(x-y)=2f\left(x\right).f\left(y\right)$.