First slide

Functions (XII)

Question

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

Moderate

A

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

{0, 1}
C

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

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

Solution

$x \in R \Rightarrow \exists n \in Z n \leq x < n + 1$
$∍ [x] = n \Rightarrow 2 n \leq 2 x < 2 n + 2 \Rightarrow [2 x] = 2 n$
or
$2 n + 1 \Rightarrow [2 x] = 2 [x]$
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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