Question

$f\left(x\right)=\frac{\cos x}{\left[\frac{2x}{\pi }\right]+\frac{1}{2}}$, where $x$ is not an integral multiple of $\pi$ and [.] denotes the greatest integer function, is

• A. an odd function
• B. an even function
• C. neither odd nor even
• D. none of these

Answer 1

The correct option is A an odd function

$f\left(x\right)=\frac{cosx}{\left[\frac{2x}{\pi }\right]+\frac{1}{2}}$

$f(-x)=\frac{cos(-x)}{\left[\frac{-2x}{\pi }\right]+\frac{1}{2}}$

$=\frac{cosx}{-1-\left[\frac{2x}{\pi }\right]+\frac{1}{2}}$

$=\frac{cosx}{-\frac{1}{2}-\left[\frac{2x}{\pi }\right]}$

$=\frac{-cosx}{\frac{1}{2}+\left[\frac{2x}{\pi }\right]}$

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

Therefore

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

$f\left(x\right)+f(-x)=0$

Hence it is an odd function.