Question

Let $f:R\to [-1,1]$ be defined by $f\left(x\right)=\frac{\sin \left(\right[x\left]\pi \right)}{x^2+1},$ where $[.]$ represents the greatest integer function. Then $f$ is

• A. one-one and into function
• B. many-one and onto function
• C. bijective function
• D. many-one and into function

Answer 1

The correct option is D many-one and into function

$\sin \left[x\right]\pi =0$ as $\left[x\right]\in Z$
$\therefore f\left(x\right)=0\forall x\in R\Rightarrow f$ is not one-one.
Also range contains only one element, i.e., $\left\{0\right\}$
$\Rightarrow f$ is also not onto.
$\therefore f$ is many-one and into function.