Question

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

• A. $f\left(x\right)$ is an even function
• B. $f\left(x\right)$ is an odd function
• C. $f\left(x\right)$ is neither even nor odd
• D. None of these

Answer 1

The correct option is B $f\left(x\right)$ is an odd function

$f\left(x\right)=\frac{\cos x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}$
$f(-x)=\frac{\cos (-x)}{\left[\frac{-x}{\pi }\right]+\frac{1}{2}}=\frac{\cos x}{-\left[\frac{x}{\pi }\right]-1+\frac{1}{2}}=-\frac{\cos x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}=-f\left(x\right)$
Hence, $f\left(x\right)$ is an odd function.