Question

Given the function $f\left(x\right)=\frac{[3^x+3^{(-x)}]}{2},thenf(x+y)+f(x-y)=?$

• A. $f\left(x\right)+f\left(y\right)$
• B. $f\left(x\right)f\left(y\right)$
• C. $\frac{f\left(x\right)}{f\left(y\right)}$
• D. $\left(\frac{1}{2}\right)f\left(x\right)\times f\left(y\right)$

Answer 1

The correct option is D

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

Explanation for the correct option::

Let

$$\begin{gathered} \Rightarrow f\left(x\right)=\frac{\left[3^x+3^{-x}\right]}{2} \\ \Rightarrow f(x+y)+f(x–y)=\frac{[3^{x+y}+3^{-x-y}]}{2}+\frac{[3^{x-y}+3^{-x+y}]}{2} \\ =\left(\frac{1}{2}\right)\frac{\left[3^x+3^{-x}\right]}{2}\times \frac{[3^y+3^{-y}]}{2} \end{gathered}$$

$\Rightarrow$$\mathit{f}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathbf{+}\mathit{f}\mathbf{(}\mathit{x}\mathbf{–}\mathit{y}\mathbf{)}\mathbf{=}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\right)}\mathbf{}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{\times }\mathit{f}\mathbf{\left(}\mathit{y}\mathbf{\right)}$

Hence, option(D) is correct.