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Functions: Denotes Integral Part Domain Function

JEE Maths question with a full step-by-step solution.

Question
If [x][x] denotes the integral part of xx, then domain of the function f(x)=3−x(x−1)(x−2)(x−3)+sin⁡−1 ⁣[3x−22]f(x)=\dfrac{\sqrt{3-x}}{(x-1)(x-2)(x-3)}+\sin^{-1}\!\left[\dfrac{3x-2}{2}\right] is:
A[0,2)[0,2)
B[0,2)−{1}[0,2)-\{1\}correct
C(−∞,3)−{1,2}(-\infty,3)-\{1,2\}
DNone of these
Solution
Condition (i): sin⁡−1\sin^{-1} requires [3x−22]∈[−1,0,1]\left[\dfrac{3x-2}{2}\right]\in[-1,0,1]:
−1≤3x−22<2  ⇒  −2≤3x−2<4  ⇒  0≤3x<6  ⇒  0≤x<2…(1)-1\le\dfrac{3x-2}{2}<2\;\Rightarrow\;-2\le 3x-2<4\;\Rightarrow\;0\le 3x<6\;\Rightarrow\;0\le x<2\quad\ldots(1)
 Condition (ii): 3−x\sqrt{3-x} requires 3−x≥0  ⇒  x≤33-x\ge 0\;\Rightarrow\;x\le 3 and x≠1,2,3…(2)x\neq 1,2,3\quad\ldots(2) Step 2: From (1) and (2): 0≤x<20\le x<2 and x≠1x\neq 1. Domain =[0,1)∪(1,2)=[0,2)−{1}=[0,1)\cup(1,2)=[0,2)-\{1\} Answer: (2) [0,2)−{1}[0,2)-\{1\}
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