Question

38. If f(x) and g(x) be two given function with all real numbers as their domain, then h(x) = {f(x)+f(-x)}{g(x)-g(-x)} is

1. Even

2. Odd

3. Even as well as odd

4. None

Answer 1

Option (2) is correct.

Given:

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

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

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

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

$h(-x)=-h\left(x\right)$

Hence, $h\left(x\right)$ is odd function