Question

найдите область определения функции
1) f(x)=logx-1 ( x/9-x²)
2)f(x)=log3-x ((x²-4)/x)​

Answer 1

Ответ:

$1)\ \ f(x)=log_{x-1}\Big(\dfrac{x}{9-x^2}\Big)\\\\\\OOF:\ \left\{\begin{array}{l}x-1>0\\x-1\ne 1\\\dfrac{x}{9-x^2}>0\end{array}\right\ \ \left\{\begin{array}{l}x>1\\x\ne 2\\\dfrac{x}{(3-x)(3+x)}>0\end{array}\right\ \ \left\{\begin{array}{l}x>1\\x\ne 2\\\dfrac{x}{(x-3)(x+3)}<0\end{array}\right$

    $\left\{\begin{array}{l}x\in (\ 1\, ;\, 2\ )\cup (\ 2\ ;+\infty \, )\\x\in (\, -\infty \, ;-3\, )\cup (\ 0\ ;\ 3\ )\end{array}\right\ \ \ \Rightarrow \ \ x\in (\ 1\ ;\ 2\ )\cup (\ 2\ ;\ 3\ )$

$2)\ \ f(x)=log_{3-x}\, \dfrac{x^2-4}{x}\\\\\\OOF:\ \ \left\{\begin{array}{l}3-x>0\\3-x\ne 1\\\dfrac{(x-2)(x+2)}{x}>0\end{array}\right\ \ \left\{\begin{array}{l}x<3\\x\ne 2\\x\in (-2\ ;\ 0\, )\cup (\, 2;+\infty )\end{array}\right\ \ \ \Rightarrow \\\\\\x\in (-2\, ;\, 0\, )\cup (\ 2\ ;\ 3\ )$