(B) Given $f(x) = [2x] - 2[x]$ for all $x \in R$. Case $1$: If $x$ is an integer, let $x = n$ where $n \in Z$. Then $f(n) = [2n] - 2[n] = 2n - 2n = 0$

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(B) Given $f(x) = [2x] - 2[x]$ for all $x \in R$. Case $1$: If $x$ is an integer, let $x = n$ where $n \in Z$. Then $f(n) = [2n] - 2[n] = 2n - 2n = 0$. Case $2$: If $x$ is not an integer, let $x = n + f$ where $n \in Z$ and $0 Then $f(x) = [2(n + f)] - 2[n + f] = [2n + 2f] - 2n = 2n + [2f] - 2n = [2f]$. Since $0 If $0 If $0.5 \leq f Thus, the range of $f(x)$ is $\{0, 1\}$.

Class 12 Mathematics Relations and Functions Domain and Range

Easy

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

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A
$\{x \in R: 0 \leq x \leq 1\}$
B
$\{0, 1\}$
C
$\{x \in R: x > 0\}$
D
$\{x \in R: x \leq 0\}$

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

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