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Консультация № 197045

11.11.2019, 15:35

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11.11.2019, 22:52

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Roman Chaplinsky / Химик CH

Модератор
156417
2175

12.11.2019, 00:26

общий

это ответ

Здравствуйте, svrvsvrv!
Данное выражение проще исследовать, начиная с внешней функции
1) Подкоренное выражение должно быть неотрицательно
log_5-x(x²-9)[$8805$]0

2) Из свойств логарифма это неравенство разбивается на 2 случая в зависимости от допустимых значений показателя
2а) показатель больше 1
5-x>1
x<4
тогда выражение под логарифмом должно быть не меньше 1 (условия области определения логарифма при этом выполняются автоматически)
x²-9[$8805$]1
x²[$8805$]10
x[$8712$](-[$8734$]; -[$8730$]10][$8746$][[$8730$]10; [$8734$])

учитывая условие x<4
x[$8712$](-[$8734$]; -[$8730$]10][$8746$][[$8730$]10; 4)

2б) показатель меньше 1 - при этом он не может быть равен 1 и должен быть положительным
0<5-x<1
x[$8712$](4; 5)
в этом случае выражение под логарифмом должно быть не больше 1, но оно также должно быть положительным
0<x²-9[$8804$]1
Впрочем, при x>4 мы автоматически получаем x²-9>4²-9=7, то есть эти условия несовместны (x[$8712$][$8709$])

таким образом, область определения, являющаяся объединением условий 2а и 2б, фактически совпадает с найденной в пункте 2а, то есть
x[$8712$](-[$8734$]; -[$8730$]10][$8746$][[$8730$]10; 4)

5

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