If $[.]$ denotes GIF, then solve for $x :$

(i) $[x] = 3$

(ii) $[2 x - 1] = 3$

x \in (3, 4]

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x \in [3, 4)

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x \in [4, 5)

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x \in [2, 2.5)

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Solution

The correct option is D $x \in [2, 2.5)$
(i) $[x] = 3$
We know $[x] \leq x < [x] + 1$
$\Rightarrow 3 \leq x < 3 + 1$
$\Rightarrow 3 \leq x < 4$
$\Rightarrow x \in [3, 4)$

(ii) $[2 x - 1] = 3$
$\Rightarrow [2 x - 1] \leq 2 x - 1 < [2 x - 1] + 1$
$\Rightarrow 3 \leq 2 x - 1 < 3 + 1$
$\Rightarrow 4 \leq 2 x < 5$
$\Rightarrow 2 \leq x < \frac{5}{2}$
$\Rightarrow x \in [2, 2.5)$

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