AP EAMCET · Maths · Functions
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 \(x\) )
- A Z, the set of all integers
- B N, the set of all natural numbers
- C R. the set of all real numbers
- D \(\{0,1\}\)
Answer & Solution
Correct Answer
(D) \(\{0,1\}\)
Step-by-step Solution
Detailed explanation
\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, &…
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