Question

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

(a) (−∞, −3] ∪ (2, 5)
(b) (−∞, −3) ∪ (2, 5)
(c) (−∞, −3) ∪ [2, 5]
(d) None of these

Answer 1

(a) (−∞, −3] ∪ (2, 5)

$f\left(x\right)=\sqrt{\frac{x+3}{\left(2-x\right)\left(x-5\right)}}$
$$\begin{gathered} \mathrm{For}\mathrm{f}\left(\mathrm{x}\right)\mathrm{to}\mathrm{be}\mathrm{defined}, \\ \left(2-x\right)\left(x-5\right)\ne 0 \\ \Rightarrow x\ne 2,5....\left(1\right) \\ \mathrm{Also},\frac{\left(x+3\right)}{\left(2-x\right)\left(x-5\right)}\ge 0 \\ \Rightarrow \frac{\left(x+3\right)\left(2-x\right)\left(x-5\right)}{{\left(2-x\right)}^2{\left(x-5\right)}^2}\ge 0 \\ \Rightarrow \left(x+3\right)\left(x-2\right)\left(x-5\right)\le 0 \\ \Rightarrow x\in (-\infty ,-3]\cup (2,5)....\left(2\right) \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right), \\ x\in (-\infty ,-3]\cup (2,5) \end{gathered}$$