The domain of definition of f(x)
Question:

The domain of definition of $f(x)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$ is

(a) (−∞, −3] ∪ (2, 5)

(b) (−∞, −3) ∪ (2, 5)

(c) (−∞, −3) ∪ [2, 5]

(d) None of these

Solution:

(a) (−∞, −3] ∪ (2, 5)

$f(x)=\sqrt{\frac{x+3}{(2-x)(x-5)}}$

For $\mathrm{f}(\mathrm{x})$ to be defined,

$(2-x)(x-5) \neq 0$

$\Rightarrow x \neq 2,5$    ….(1)

Also, $\frac{(x+3)}{(2-x)(x-5)} \geq 0$

$\Rightarrow \frac{(x+3)(2-x)(x-5)}{(2-x)^{2}(x-5)^{2}} \geq 0$

 

$\Rightarrow(x+3)(x-2)(x-5) \leq 0$

$\Rightarrow x \in(-\infty,-3] \cup(2,5) \quad \ldots(2)$

From $(1)$ and $(2)$

$x \in(-\infty,-3] \cup(2,5)$

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