Question

The domain of the function \[f\left( x \right) = \sqrt{\frac{\left( x + 1 \right) \left( x - 3 \right)}{x - 2}}\] is

  

Options

• (a) [−1, 2) ∪ [3, ∞)

• (b) (−1, 2) ∪ [3, ∞)

• (c) [−1, 2] ∪ [3, ∞)

• (d) None of these

   

Answer 1

(a) [−1, 2) ∪ [3, ∞) 

\[f\left( x \right) = \sqrt{\frac{\left( x + 1 \right) \left( x - 3 \right)}{x - 2}}\]

\[\text{ For f(x) to be defined, }  \]
\[(x - 2) \neq 0\]
\[ \Rightarrow x \neq 2 . . . (1)\]
\[\text{ Also, } \]
\[\frac{(x + 1)(x - 3)}{(x - 2)} \geq 0\]
\[ \Rightarrow \frac{(x + 1)(x - 3)(x - 2)}{(x - 2 )^2} \geq 0\]
\[ \Rightarrow (x + 1)(x - 3)(x - 2) \geq 0\]
\[ \Rightarrow x \in [ - 1, 2) \cup [3, \infty ) . . . . . (2)\]
\[\text{ From (1) and (2),} \]
\[x \in [ - 1, 2) \cup [3, \infty ) \]