First slide

Functions (XII)

Question

The domain of the function $f (x) = \log_{[x + \frac{1}{2}]} |x^{2} - 5 x + 6| is$

Moderate

A

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

$[\frac{3}{2}, \infty)$
C

$[\frac{1}{2}, \infty)$
D

None of these

Solution

f (x) is defined if
$x^{2} - 5 x + 6 \neq 0, [x + \frac{1}{2}] > 0, [x + \frac{1}{2}] \neq 1 x^{2} - 5 x + 6 \neq 0 \Rightarrow (x - 2) (x - 3) \neq 0 \Rightarrow x \neq 2, 3 . . . . . . (1) \begin{array}{l} [x + \frac{1}{2}] > 0 \Rightarrow x \geq \frac{1}{2} . . . . . . (2) \\ [x + \frac{1}{2}] \neq 1 \Rightarrow x \notin [\frac{1}{2}, \frac{3}{2}) . . . . . . (3) \end{array}$
From Eq. (1), (2) and (3), we get domain of f
$= [\frac{3}{2}, 2) \cup (2, 3) \cup (3, \infty)$

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