Let [x] denote the greatest integer ≤x, where x∈R if the domain of the real valued function f(x)=|[x]|−2|[x]|−3 is (−∞, a)∪[b, c)∪[4, ∞), a<b<c, then the value of a+b+c is:

f(x)= |[x]|−2|[x]|−3 is defined iff|[x]|−2|[x]|−3≥0case(i): |[x]|−2≥0 and |[x]|−3> 0|[x]|≥2,|[x]|>3[x]≤−2 or [x]≥2,[x ] 3x≤−2 or x≥2,x< −3 or x≥4∴ x∈(−∞, −3) ∪ [4, ∞)−−−−(1)case(ii): |[x]|−2≤0 and |[x]|−3< 0|[x]|≤2,|[x]|<3−2≤[x]≤2,−3<[x]<3∴ x∈[−2, 3)−−−−(2)from (1) &(2)x∈(−∞, −3) ∪[−2, 3)∪[4, ∞)∴ a=−3, b=−2, c=3∴ a+b+c=−3−2+3=−2

Let [x] denote the greatest integer ≤x, where x∈R if the domain of the real valued function f(x)=|[x]|−2|[x]|−3 is (−∞, a)∪[b, c)∪[4, ∞), a<b<c, then the value of a+b+c is:

f(x)= |[x]|−2|[x]|−3 is defined iff|[x]|−2|[x]|−3≥0case(i): |[x]|−2≥0 and |[x]|−3> 0|[x]|≥2,|[x]|>3[x]≤−2 or [x]≥2,[x ] 3x≤−2 or x≥2,x< −3 or x≥4∴ x∈(−∞, −3) ∪ [4, ∞)−−−−(1)case(ii): |[x]|−2≤0 and |[x]|−3< 0|[x]|≤2,|[x]|<3−2≤[x]≤2,−3<[x]<3∴ x∈[−2, 3)−−−−(2)from (1) &(2)x∈(−∞, −3) ∪[−2, 3)∪[4, ∞)∴ a=−3, b=−2, c=3∴ a+b+c=−3−2+3=−2

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$\begin{array}{l} L e t [x] d e n o t e t h e g r e a t e s t int e g e r \leq x, w h e r e x \in R i f t h e d o m a i n o f t h e r e a l v a l u e d f u n c t i o n \\ f (x) = \sqrt{\frac{| [x] | - 2}{| [x] | - 3}} i s (- \infty, a) \cup [b, c) \cup [4, \infty), a < b < c, t h e n t h e v a l u e o f a + b + c i s : \end{array}$

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Detailed Solution

$\begin{array}{l} f (x) = \sqrt{\frac{| [x] | - 2}{| [x] | - 3}} i s d e f i n e d i f f \\ \frac{| [x] | - 2}{| [x] | - 3} \geq 0 \\ c a s e (i) : | [x] | - 2 \geq 0 a n d | [x] | - 3 > 0 \\ | [x] | \geq 2, | [x] | > 3 \\ [x] \leq - 2 o r [x] \geq 2, [x] < - 3 o r [x] > 3 \\ x \leq - 2 o r x \geq 2, x < - 3 o r x \geq 4 \\ ∴ x \in (- \infty, - 3) \cup [4, \infty) - - - - (1) \end{array}$

$\begin{array}{l} c a s e (i i) : | [x] | - 2 \leq 0 a n d | [x] | - 3 < 0 \\ | [x] | \leq 2, | [x] | < 3 \\ - 2 \leq [x] \leq 2, - 3 < [x] < 3 \\ ∴ x \in [- 2, 3) - - - - (2) \\ f r o m (1) & (2) \\ x \in (- \infty, - 3) \cup [- 2, 3) \cup [4, \infty) \\ ∴ a = - 3, b = - 2, c = 3 \\ ∴ a + b + c = - 3 - 2 + 3 = - 2 \end{array}$