Question

$f\left(x\right)=\frac{\sin \left[x\right]\pi }{x^2+x+1}$ where $[.]$ denotes the greatest integer function then $f\left(x\right)$ is

• A. one - one function
• B. one - one and in to function
• C. constant fuction
• D. onto function

Answer 1

The correct option is C constant fuction

$f\left(x\right)=\frac{\sin \left[x\right]\pi }{x^2+x+1}=\frac{p\left(x\right)}{q\left(x\right)}$

$q\left(x\right)=2^2+x+1$

$D=b^2-4ac=-1-4\times 1\times 1=-5<0$

So it have only imaginary root.

$p\left(x\right)=\sin \left[x\right]\pi$

$\left[x\right]\Rightarrow$ shows any integer value

So, $\sin \left[x\right]\pi =0$ for all $x\in R[\because \sin n\pi =0.n\in integer]$

$\sin \left[x\right]\pi =0$

$f\left(x\right)=\frac{0}{n^2+n+1}=0$ Which is constant function.