Question

If $f:R\to [-1,1]$ where $f\left(x\right)=\sin \left(\frac{\pi }{2}\right[x\left]\right)$ $\left(\right[.]$ denotes greatest integer function$)$, then $f\left(x\right)$ is:

• A. an onto function
• B. an into function
• C. a periodic function
• D. many-one function

Answer 1

Answer: (B), (C), (D)

many-one function

We know,
$\left[x\right]=\{\cdots ,-1,0,1,2,3,\cdots \}$
When $\left[x\right]$ is even integer, then
$f\left(x\right)=0$
When $\left[x\right]$ is odd integer, then
$f\left(x\right)=\pm 1$
Clearly range of $f\left(x\right)$ is $\{-1,0,1\}$
As range $\ne$ codomain, so $f\left(x\right)$ is into function.

For $\forall x\in [0,1),\left[x\right]=0\Rightarrow f\left(x\right)=0$, so $f\left(x\right)$ is a many-one function.

When
$x\in [1,2)\Rightarrow \left[x\right]=1\Rightarrow f\left(x\right)=1x\in [2,3)\Rightarrow \left[x\right]=2\Rightarrow f\left(x\right)=0x\in [3,4)\Rightarrow \left[x\right]=3\Rightarrow f\left(x\right)=-1x\in [4,5)\Rightarrow \left[x\right]=4\Rightarrow f\left(x\right)=0x\in [5,6)\Rightarrow \left[x\right]=5\Rightarrow f\left(x\right)=1x\in [6,7)\Rightarrow \left[x\right]=6\Rightarrow f\left(x\right)=0x\in [7,8)\Rightarrow \left[x\right]=7\Rightarrow f\left(x\right)=-1x\in [8,9)\Rightarrow \left[x\right]=8\Rightarrow f\left(x\right)=0$
Hence $f\left(x\right)$ is a periodic function with period $4.$