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If $[.]$ denotes the greatest integer function then domain of the real valued function $f (x) = \log_{[x+1/2]} |x^{2} + x - 6|$ is

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$(\frac{1}{2}, 1) \cup (1, \infty)$

$[\frac{3}{2}, 2) \cup (2, \infty)$

$[0, \frac{3}{2}] \cup (2, \infty)$

$(0, 1] \cup (\frac{3}{2}, \infty)$

answer is C.

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Detailed Solution

We must have $x^{2} + x - 6 \neq 0$ and $[x + \frac{1}{2}] > 0, [x + \frac{1}{2}] \neq 1$
$\Rightarrow x \neq - 3, 2$ and $[x + \frac{1}{2}] \geq 2$
$\Rightarrow x \geq \frac{3}{2}$
$\Rightarrow x \in [\frac{3}{2}, 2) \cup (2, \infty)$

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