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Question
The domain of definition of \[f\left( x \right) = \sqrt{\frac{x + 3}{\left( 2 - x \right) \left( x - 5 \right)}}\] is
Options
(a) (−∞, −3] ∪ (2, 5)
(b) (−∞, −3) ∪ (2, 5)
(c) (−∞, −3) ∪ [2, 5]
(d) None of these
Solution
(a) (−∞, −3] ∪ (2, 5)
\[ \text{ For f(x) to be defined ,} \]
\[\left( 2 - x \right)\left( x - 5 \right) \neq 0\]
\[ \Rightarrow x \neq 2, 5 . . . . (1)\]
\[\text{ Also } , \frac{\left( x + 3 \right)}{\left( 2 - x \right)\left( x - 5 \right)} \geq 0\]
\[ \Rightarrow \frac{\left( x + 3 \right)\left( 2 - x \right)\left( x - 5 \right)}{\left( 2 - x \right)^2 \left( x - 5 \right)^2} \geq 0\]
\[ \Rightarrow \left( x + 3 \right)\left( x - 2 \right)\left( x - 5 \right) \leq 0\]
\[ \Rightarrow x \in ( - \infty , - 3] \cup (2, 5) . . . . (2)\]
\[\text{ From (1) and (2),} \]
\[x \in ( - \infty , - 3] \cup (2, 5)\]
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