tex] A)( -1 ; 2,2 ] B) [ 2,2 ; 3 ) C) (-∞;3) D) (-1;3)...

July 2022 1 11 Report

Найдите область определения функции:
[tex]f(x)= \sqrt{log_{0,5} \frac{x+1}{12-4x} } [/tex]

A)( -1 ; 2,2 ] B) [ 2,2 ; 3 ) C) (-∞;3) D) (-1;3)

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NNNLLL54

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$f(x)=\sqrt{log_{0,5}\, \frac{x+1}{12-4x} }\\\\OOF:\; \; \left \{ {{log_{0,5} \frac{x+1}{12-4x} \geq 0 } \atop { \frac{x+1}{12-4x}\ \textgreater \ 0 }} \right. \\\\a)\; \; log_{0,5}\, \frac{x+1}{12-4x}=-log_2\, \frac{x+1}{12-4x} \geq 0\\\\log_2\, \frac{x+1}{12-4x} \leq 0\; ,\; \; \; log_2\, \frac{x+1}{12-4x} \leq log_21\; \; \Rightarrow \; \; \frac{x+1}{12-4x} \leq 1\\\\ \frac{x+1}{12-4x} -1\leq 0\; ,\; \; \; \frac{x+1-12+4x}{12-4x} \leq 0\; ,\; \; \frac{5x-11}{4(x-3)} \geq 0$

$Znaki:\; \; \; +++[\, 2,2\, ]---(3)+++$

$x\in (-\infty ;\; 2,2\, ]\cup (3,+\infty )\\\\b)\; \; \frac{x+1}{12-4x}\ \textgreater \ 0\; ,\; \; \frac{x+1}{-4(x-3)}\ \textgreater \ 0\; ,\; \; \frac{x+1}{x-3}\ \textless \ 0\\\\Znaki:\; \; \; +++(-1)---(3)+++\\\\x\in (-1,3\, )\\\\c)\; \; \left \{ {{x\in (-\infty ;\, 2,2\, )\cup (3,+\infty )} \atop {x\in (-1,3)}} \right. \; \; \; \Rightarrow \; \; \; \boxed {\; x\in (-1\, ;\, 2,2\; ]\; }$

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