The domain of fx=x−3x+3+2−x2+x is

fx=x−3x+3+2−x2+x is defined iffx−3x+3≥0, and 2−x2+x≥0⇒ x+3x−3≥0 and 2−x2+x≥0⇒ x≤−3 or x≥3andx−2x+2≤0⇒ x∈−2, 2

The domain of fx=x−3x+3+2−x2+x is

fx=x−3x+3+2−x2+x is defined iffx−3x+3≥0, and 2−x2+x≥0⇒ x+3x−3≥0 and 2−x2+x≥0⇒ x≤−3 or x≥3andx−2x+2≤0⇒ x∈−2, 2

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$T h e d o m a i n o f f (x) = \sqrt{\frac{x - 3}{x + 3}} + \sqrt{\frac{2 - x}{2 + x}} i s$

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$(2, 3)$

$(- \infty, 2) \cup (3, \infty)$

$ϕ$

answer is D.

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Detailed Solution

$\begin{array}{l} f (x) = \sqrt{\frac{x - 3}{x + 3}} + \sqrt{\frac{2 - x}{2 + x}} i s d e f i n e d i f f \\ \frac{x - 3}{x + 3} \geq 0, a n d \frac{2 - x}{2 + x} \geq 0 \\ \Rightarrow (x + 3) (x - 3) \geq 0 a n d (2 - x) (2 + x) \geq 0 \\ \Rightarrow x \leq - 3 o r x \geq 3 a n d (x - 2) (x + 2) \leq 0 \Rightarrow x \in [- 2, 2] \end{array}$