For \(f(x)=\frac{\sin \pi[x]}{1+[x]}+\frac{x}{2+3 x}\), where \([x]\) denotes the greatest integer function,…
For \(f(x)=\frac{\sin \pi[x]}{1+[x]}+\frac{x}{2+3 x}\), where \([x]\) denotes the greatest integer function, the domain and range in \(R\) are respectively
\(R-\left\{-1, \frac{-2}{3}\right\}\) and \(R-\left\{\frac{1}{3}\right\}\)
\(\mathrm{R}-\left\{-1, \frac{-2}{3}\right\}\) and \([-1,1]\)
$R-(-1,0)$ and $R-\left\{\frac{1}{3}\right\}$
$R - (-1,0)$ and $(-1,1)$
Solution
As $[x]=-1$ when
$x \in[-1,0)$. This makes the denominator of the first part $1+[x]=0$. Hence,
the interval $[-1,0)$ must be excluded from the domain set.
$\therefore \mathrm{D}(f)=R-[-1,0)$
Also at $x=0$ (which is part of the domain), the value of the function is zero.
i.e. $f(0)=0$
So option (d) is correct.