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 the above

Answer 1

The correct option is A an odd function

$f(-x)=\frac{\cos (-x)}{[-\frac{2x}{\pi }]+\frac{1}{2}}\Rightarrow f(-x)=\frac{\cos x}{-1-\left[\frac{2x}{\pi }\right]+\frac{1}{2}}$
(as $x$ is not an integral multiple of $\pi \Rightarrow \frac{2x}{\pi }$ will not be an integer)
$\Rightarrow f(-x)=-\frac{\cos x}{\left[\frac{2x}{\pi }\right]+\frac{1}{2}}=-f\left(x\right)$
$\Rightarrow f\left(x\right)$ is an odd function.