Question

Let $f\left(x\right)$ and $g\left(x\right)$ are two function of $x$ such that $f\left(x\right)+g\left(x\right)=e^x$ and $f\left(x\right)-g\left(x\right)=e^{-x}$ then

• A. $f\left(x\right)$ is an odd function
• B. $g\left(x\right)$ is an odd function
• C. $f\left(x\right)$ is an even function
• D. $g\left(x\right)$ is an even function

Answer 1

Answer: (B), (C)

$g\left(x\right)$ is an odd function
$f\left(x\right)$ is an even function

$f\left(x\right)+g\left(x\right)=e^x$...(i)
$f\left(x\right)-g\left(x\right)=e^{-x}$ ...(ii)
Adding both the equations, we get
$f\left(x\right)=\frac{e^x+e^{-x}}{2}$
$f(-x)=\frac{e^{-x}+e^x}{2}$
$=f\left(x\right)$
Hence $f\left(x\right)$ is an even function.
Subtracting ii from i, we get
$g\left(x\right)=\frac{e^x-e^{-x}}{2}$
$g(-x)=\frac{e^{-x}-e^x}{2}$
$=\frac{-(e^x-e^{-x})}{2}$
$=-g\left(x\right)$
Hence $g\left(x\right)$ is an odd function.