Question

Whether the given function is even or odd

${\left[\frac{f\left(x\right)-f(-x)}{g\left(x\right)+g(-x)}\right]}^n$.

Answer 1

Let, $F\left(x\right)={\left[\frac{f\left(x\right)-f(-x)}{g\left(x\right)+g(-x)}\right]}^n$

Now, $F(-x)={\left[\frac{f(-x)-f\left(x\right)}{g(-x)+g\left(x\right)}\right]}^n={(-1)}^n{\left[\frac{f\left(x\right)-f(-x)}{g\left(x\right)+g(-x)}\right]}^n$

$\Rightarrow F(-x)=(-1{)}^{n}F(x)$

Therefore, if $n$ is even then F is even and if $n$€ is odd then F is odd.