Question

The domain of the function $f\left(x\right)=\sqrt{\frac{\left(x+1\right)\left(x-3\right)}{x-2}}$is
(a) [−1, 2) ∪ [3, ∞)
(b) (−1, 2) ∪ [3, ∞)
(c) [−1, 2] ∪ [3, ∞)
(d) None of these

Answer 1

(a) [−1, 2) ∪ [3, ∞)

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