If f:R→R is defined by f(x)=2x−2[x] for x∈R, where [x] is the greatest integer not exceeding x, then the range of f is

x∈R⇒∃n∈Zn≤x<n+1∍[x]=n⇒2n≤2x<2n+2⇒[2x]=2nor2n+1⇒[2x]=2[x]or 2[x]+1⇒[2x]=2[x]or 1⇒f(x)=0 or 1⇒range={0,1}

If f:R→R is defined by f(x)=2x−2[x] for x∈R, where [x] is the greatest integer not exceeding x, then the range of f is

x∈R⇒∃n∈Zn≤x<n+1∍[x]=n⇒2n≤2x<2n+2⇒[2x]=2nor2n+1⇒[2x]=2[x]or 2[x]+1⇒[2x]=2[x]or 1⇒f(x)=0 or 1⇒range={0,1}

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If $f : R \to R$ is defined by $f (x) = [2 x] - 2 [x]$ for $x \in R,$ where [x] is the greatest integer not exceeding x, then the range of f is

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$\{x \in R; 0 \leq x \leq 1\}$

{0, 1}

$\{x \in R; x > 0\}$

$\{x \in R; x < 0\}$

answer is B.

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Detailed Solution

$x \in R \Rightarrow \exists n \in Z n \leq x < n + 1$

$∍ [x] = n \Rightarrow 2 n \leq 2 x < 2 n + 2 \Rightarrow [2 x] = 2 n$

$2 n + 1 \Rightarrow [2 x] = 2 [x]$

or $2 [x] + 1 \Rightarrow [2 x] = 2 [x]$

or $1 \Rightarrow f (x) = 0 o r 1 \Rightarrow r a n g e = {0, 1}$

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If f:R→R is defined by f(x)=2x−2[x] for x∈R, where [x] is the greatest integer not exceeding x, then the range of f is

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If $f : R \to R$ is defined by $f (x) = [2 x] - 2 [x]$ for $x \in R,$ where [x] is the greatest integer not exceeding x, then the range of f is