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16 Jul 2024, 07:00
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B
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Difficulty:
95%
(hard)
Question Stats:
44% (02:20) correct
56%
(02:31)
wrong
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The domain of the function \(f(x) = \frac{\sqrt{|x| - 2}}{\sqrt{\sqrt{x + 9}- \sqrt{3 - x}}}\) is the set of real numbers x such that:
A. \(-3 < x ≤ -2\) or \(2 ≤ x < 3\)
B. \(-3 < x ≤ -2\) or \(2 ≤ x ≤ 3\)
C. \(-3 ≤ x ≤ -2\) or \(2 ≤ x ≤ 3\)
D. \(-9 ≤ x < -3\) or \(-3 < x ≤ -2\) or \(2 ≤ x ≤ 3\)
E. \(-9 ≤ x ≤ -2\) or \(2 ≤ x ≤ 3\)
A. \(-3 < x ≤ -2\) or \(2 ≤ x < 3\)
B. \(-3 < x ≤ -2\) or \(2 ≤ x ≤ 3\)
C. \(-3 ≤ x ≤ -2\) or \(2 ≤ x ≤ 3\)
D. \(-9 ≤ x < -3\) or \(-3 < x ≤ -2\) or \(2 ≤ x ≤ 3\)
E. \(-9 ≤ x ≤ -2\) or \(2 ≤ x ≤ 3\)
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16 Jul 2024, 08:30
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Bunuel
GMAT Club Official Explanation:
Given that the expression contains a fraction and square roots, the domain of the function will be defined when the square roots are greater than or equal to 0 (since even roots are not defined for negative numbers) and the denominator of the fraction is not equal to 0 (since division by 0 is not allowed). Thus, the following conditions must be satisfied simultaneously:
1. The expression inside the square root in the numerator, |x| - 2, must be non-negative:
|x| - 2 ≥ 0
|x| ≥ 2
This means x ≤ -2 or x ≥ 2.
--------------------------(-2)--------(2)-------
2. The individual expressions inside the square roots in the denominator, x + 9, and 3 - x must be non-negative:
x + 9 ≥ 0 and 3 - x ≥ 0
x ≥ -9 and x ≤ 3
This means -9 ≤ x ≤ 3.
--(-9)-------------------------------------(3)--
3. The whole expression inside the square roots in the denominator must be positive:
\(\sqrt{x + 9} - \sqrt{3 - x} > 0\)
\(\sqrt{x + 9} > \sqrt{3 - x}\)
\(x + 9 > 3 - x\)
\(2x > -6\)
\(x > -3\)
-------------------(-3)---------------------------
The overlap of these ranges is \(-3 < x ≤ -2\) and \(2 ≤ x ≤ 3\).
--(-9)------------(-3)--(-2)--------(2)--(3)--
Answer: B.
General Discussion
16 Jul 2024, 07:37
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Bunuel
IMO B
Making whatever is inside the root as non-negative
Numerator: |x|>=2 => x>=2, x<=-2
Denominator: \sqrt{ x+9 } -\sqrt{3 -x } > 0
x >-3
\sqrt{3 -x }3-x >=0
x<=3
