Q. If $f:R \rightarrow R$ is defined by $f(x) = \lbrack 2x\rbrack - 2\lbrack x\rbrack$ for all $x \in R$, where $\lbrack x\rbrack$ is the greatest integer not exceeding $x$, then the range of $f$ is
A) $\{ x \in R:0 \leq x \leq 1\}$
B) {0, 1}
C) $\{ x \in R:x > 0\}$
D) $\{ x \in R:x \leq 0\}$
Explanation
(b)
Given, $f(x) = \lbrack 2x\rbrack - 2\lbrack x\rbrack,\ \forall\ x \in R$
If $x$ is an integer, then
$f(x) = 0$
And if $x$ is an integer, then
$f(x)$ is either 1 or 0.
$\therefore$ Range of $f(x) = \{ 0,\ 1\}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.