Q. If $f:R \rightarrow R$ is defined by $f(x) = x - \lbrack x\rbrack - \frac{1}{2}$ for $x \in R$, where $\lbrack x\rbrack$ is the greatest integer not exceeding $x$, then $\left\{ x \in R:f(x) = \frac{1}{2} \right\}$ is equal to
A) $Z$, the set of all integers
B) $N$, the set of all natural numbers
C) $\phi$, the empty set
D) $R$
Explanation
(c)
Since, $f(x) = x - \lbrack x\rbrack - \frac{1}{2}$
Also, $f(x) = \frac{1}{2}$
$\therefore\ \ \ \frac{1}{2} = x - \lbrack x\rbrack - \frac{1}{2}$
$\Rightarrow \ \ x - \lbrack x\rbrack = 1$
$\Rightarrow \ \ \left\{ x \right\} = 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lbrack\because x = \lbrack x\rbrack + \{ x\}\rbrack$
Which is not possible.
$\therefore\ \left\{ x \in R:f(x) = \frac{1}{2} \right\}$ is an empty set.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.