First slide

Functions (XII)

Question

The domain of the function $f (x) = \frac{\sin^{- 1} (3 - x)}{l n (|x| - 2)}$ is

Moderate

A

[2, 4]
B

$\begin{array}{l} (2, 3) \cup (3, 4] \end{array}$
C

$\begin{array}{l} (2, \infty] \end{array}$
D

$(- \infty, - 3) \cup [2, \infty)$

Solution

$\begin{array}{l} f (x) - \frac{\sin^{- 1} (3 - x)}{\log (|x| - 2)} \\ Let g (x) = \sin^{- 1} (3 - x) \\ or - 1 \leq 3 - x \leq 1 \end{array}$
The domain of g(x) is [2,4].
Let $\begin{array}{l} h (x) = \log (|x| - 2) \\ i . e ., |x| - 2 > 0 or |x| > 2 \\ i . e ., x < - 2 or x > 2 \\ ∴ Domain (- \infty, - 2) \cup (2, \infty) \end{array}$
We know that $(f / g) (x) = \frac{f (x)}{g (x)} \forall x \in D_{1} \cap D_{2} - \{x \in R : g (x) = 0\}$

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