Question

Let $f\left(x\right)=\frac{(2^x+1{)}^2}{2^x}$ and $g\left(x\right)={\sin }^3{x}^3.$
If $fog$ and $gof$ are defined, then which of the following is/are correct

• A. $fog$ is an even function
• B. $fog$ is an odd function
• C. $gof$ is an odd function
• D. $gof$ is an even function

Answer 1

Answer: (A), (D)

$gof$ is an even function

$g(-x)={\sin }^3(-x{)}^3=-{\sin }^3{x}^3$
So, $g\left(x\right)$ is an odd function

$f(-x)=\frac{(2^{-x}+1{)}^2}{2^{-x}}$
$=\frac{(1+2^x{)}^2}{2^x}$
$=f\left(x\right)$
So, $f\left(x\right)$ is an even function.

As composition of functions (which are either even or odd) will be even if at least one of them is even.
So, $fog$ and $gof$ are even functions.