The domain of the function is (where [x] denotes the greatest integer less than or equal to x). [2023]

Question

The domain of the function $f (x) = \frac{1}{\sqrt{{[x]}^{2} - 3 [x] - 10}}$ is (where [x] denotes the greatest integer less than or equal to x). [2023]

Smart Achievers · Accepted Answer

(2)

${[x]}^{2} - 3 [x] - 10 > 0$

$\Rightarrow {[x]}^{2} - 5 [x] + 2 [x] - 10 > 0 \Rightarrow ([x] - 5) ([x] + 2) > 0$

$\Rightarrow [x] < - 2 or [x] > 5 \Rightarrow x < - 2 or x \geq 6$

$\Rightarrow x \in (- \infty, - 2) \cup [6, \infty)$

Solution

Q.
The domain of the function $f (x) = \frac{1}{\sqrt{{[x]}^{2} - 3 [x] - 10}}$ is (where [x] denotes the greatest integer less than or equal to x). [2023]

1 $(- \infty, - 3] \cup (5, \infty)$

2 $(- \infty, - 2) \cup [6, \infty)$

3 $(- \infty, - 3] \cup [6, \infty)$

4 $(- \infty, - 2) \cup (5, \infty)$

Ans.
(2)

${[x]}^{2} - 3 [x] - 10 > 0$

$\Rightarrow {[x]}^{2} - 5 [x] + 2 [x] - 10 > 0 \Rightarrow ([x] - 5) ([x] + 2) > 0$

$\Rightarrow [x] < - 2 or [x] > 5 \Rightarrow x < - 2 or x \geq 6$

$\Rightarrow x \in (- \infty, - 2) \cup [6, \infty)$

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Solution

Q. The domain of the function f(x)=1[x]2-3[x]-10 is (where [x] denotes the greatest integer less than or equal to x). [2023]

1 (-∞,-3]∪(5,∞)

2 (-∞,-2)∪[6,∞)

3 (-∞,-3]∪[6,∞)

4 (-∞,-2)∪(5,∞)

Ans. (2) [x]2-3[x]-10>0 ⇒[x]2-5[x]+2[x]-10>0 ⇒([x]-5)([x]+2)>0 ⇒[x]<-2 or [x]>5 ⇒x<-2 or x≥6 ⇒x∈(-∞,-2)∪[6,∞)