Question

Найти область определения функции

Answer 1

$1)quad y=sqrt[4]{ frac{125^{3x-5}}{64^{3x-5}}-(1frac{9}{16} )^{1+x}}\\OOF:; ;  frac{(5^3)^{3x-5}}{(4^3)^{3x-5}}-(frac{25}{16})^{1+x} geq 0\\(frac{5}{4})^{9x-15} geq (frac{5}{4})^{2+2x}\\frac{5}{4}>1; ; to ; ; 9x-15 geq 2+2x\\7x geq 17; ,; x geq frac{17}{7}; ,; ; x geq 2frac{3}{7}\\xin [, 2frac{3}{7}; ;; +infty )$

$2)quad y= frac{x^2-64}{25^{x-0,5}+4cdot 5^{x}-1} \\OOF:; ; (5^2)^{x-0,5}+4cdot 5^{x}-1ne 0\\5^{2x-1}+4cdot 5^{x}-1ne 0\\5^{2x}cdot 5^{-1}+4cdot 5^{x}-1ne 0\\t=5^{x}>0; ,; ; t^2cdot frac{1}{5}+4t-1ne 0; ; to \\t^2+20t-5ne 0\\D=420; ,; ; t_{1,2}=frac{-20pm sqrt{420}}{2}=-10pm sqrt{105}\\t_1approx 0,25; ;; ; t_2approx -20,25<0\\5^{x}ne -10+sqrt{105}; ; to ; ; xne log_5(-10+sqrt{105})\\xin (-infty ,-10+sqrt{105})cup (-10+sqrt{105},+infty )$