Question

If $f$ is an even function and $g$ is odd function, then the function $f\circ g$ is

• A. An even function
• B. An odd function
• C. Neither even nor odd
• D. A periodic function

Answer 1

The correct option is A An even function

$f$ is even (given)

$\therefore f(-x)=f\left(x\right)$

$g$ is odd (given)

$\therefore g(-x)=-g\left(x\right)$

So,

According to question,

$fog=f\left[g\right(x\left)\right]=f\left(y\right)$ Let $\left[g\right(x\left)\right]=y$

and also,

$fog=f\left[g\right(-x\left)\right]$ $\because g\left(x\right)$ is odd

$=f[-g(x\left)\right]$

$=f(-y)$

$=f\left(y\right)$

$\Rightarrow fog\left(x\right)=fog(-x)$

$\therefore fog$ is an even function