The domain of the function f ( x )=√ log 1 / 2( log 5([ x 2]-3)) is [Note: Where [ k ] denotes greatest integer function less than or equal to k ]

Question

The domain of the function f ( x )=√ log 1 / 2( log 5([ x 2]-3)) is [Note: Where [ k ] denotes greatest integer function less than or equal to k ] (A) [-3,-2] ∪[2,3] (B) (-3,-2] ∪[2,3) (C) (-3,-√5] ∪[√5, 3) (D) (-2 √2,-2) ∪(2,2 √2)

Tardigrade Admin · Accepted Answer

f ( x )=√ log 1 / 2 log 5([ x 2]-3) 0 ≤ log 1 / 2 log 5([ x 2]-3) ⇒ 0< log 5([ x 2]-3) ≤ 1 ⇒ 1<[x2]-3 ≤ 5 4<[ x 2] ≤ 8 5 ≤ x 2<9 x ∈(-3,-√5] ∪[√5, 3) .

Q. The domain of the function $f (x) = lo g_{1/2} (lo g_{5} ([x^{2}] - 3))$ is
[Note: Where $[k]$ denotes greatest integer function less than or equal to $k$ ]

$[- 3, - 2] \cup [2, 3]$

$(- 3, - 2] \cup [2, 3)$

$(- 3, - 5] \cup [5, 3)$

$(- 22, - 2) \cup (2, 22)$

$f (x) = lo g_{1/2} lo g_{5} ([x^{2}] - 3)$
$0 \leq lo g_{1/2} lo g_{5} ([x^{2}] - 3)$
$\Rightarrow 0 < lo g_{5} ([x^{2}] - 3) \leq 1$
$\Rightarrow 1 < [x^{2}] - 3 \leq 5$
$4 < [x^{2}] \leq 8$
$5 \leq x^{2} < 9$
$x \in (- 3, - 5] \cup [5, 3) .$

Q. The domain of the function f(x)=log1/2​(log5​([x2]−3))​ is [Note: Where [k] denotes greatest integer function less than or equal to k ]

[−3,−2]∪[2,3]

(−3,−2]∪[2,3)

(−3,−5​]∪[5​,3)

(−22​,−2)∪(2,22​)