[tex]y= 1/\sqrt{250*5^{3-x} -2*5^{x-3} }[/tex]Найдите область определения функции. Created by julia05022003. algebra-ru

tex]Найдите область определения функции...

julia05022003 @julia05022003

July 2022 1 7 Report

[tex]y= 1/\sqrt{250*5^{3-x} -2*5^{x-3} }[/tex]Найдите область определения функции

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$y=\frac{1}{\sqrt{250\cdot 5^{3-x}-2\cdot 5^{x-3}}}\\\\OOF:\; \; 250\cdot 5^{3-x}-2\cdot 5^{x-3}>0\\\\250\cdot 5^3\cdot 5^{-x}-2\cdot 5^{x}\cdot 5^{-3}>0\\\\t=5^{x}>0\; \; \to \; \; \; 5^{-x}=t^{-1}=\frac{1}{t}\; \; ,\; \; \; \frac{250\cdot 125}{t}-\frac{2\, t}{125}>0\; ,\\\\\frac{250\cdot 125\cdot 125-2t^2}{125\, t}>0\; \; ,\\\\250\cdot 125^2-2t^2=0\; ,\; 125^3-t^2=0\; \; ,\\\\t^2=125^3\; \; ,\; \; t^2=(5^3)^3\; \; ,\; \; t^2=5^9\; \; ,\; \; t=\pm \sqrt{5^9}$

$\frac{(t-\sqrt{5^9})(t+\sqrt{5^9})}{t}>0\qquad ---(-\sqrt{5^9})+++(0)---(\sqrt{5^9})+++\\\\t\in (-\sqrt{5^9},0)\cup (\sqrt{5^9},+\infty )\\\\t>0\; \; \to \; \; 5^{x}>\sqrt{5^9}\; \; ,\; \; 5^{x}>5^{4,5}\; \; ,\; \; \underline {x>4,5}\\\\\underline {\, x\in (\; 4,5\; ;+\infty )\; }$

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