If $f: R \rightarrow R$ is defined by $f(x)=[2 x]-2[x]$ for $x \in R$, then the range of $f$ is (Here $[x]$…

If $f: R \rightarrow R$ is defined by $f(x)=[2 x]-2[x]$ for $x \in R$, then the range of $f$ is (Here $[x]$ denotes the greatest integer not exceeding $x$ )
  1. Z, the set of all integers
  2. N, the set of all natural numbers
  3. R. the set of all real numbers
  4. $\{0,1\}$

Solution

$ \text { Since, } \begin{aligned} x & =[x]+\{x\} \\ \Rightarrow \quad 2 x & =2[x]+2\{x\} \\ {[2 x] } & =2[x]+(2[x]) \\ {[2 x] } & = \begin{cases}2[x]+0, & 0 < \{x\} < \frac{1}{2} \\ 2[x]+1, & \frac{1}{2} \leq\{x\} < 1\end{cases} \\ \therefore \quad[2 x]-2[x] & = \begin{cases}0, & 0 \leq\{x\} < \frac{1}{2} \\ 1, & \frac{1}{2} \leq\{x\} < 1\end{cases} \end{aligned} $ Since, $ \therefore \quad[2 x]-2[x]= \begin{cases}0, & 0 \leq\{x\} < \frac{1}{2} \\ 1, & \frac{1}{2} \leq\{x\} < 1\end{cases} $ Hence, Range of $f$ is $\{0,1\}$

Asked in: AP EAMCET 2018 (22 Apr Shift 1)

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