Question

Explicite o domínio da função

Answer 1

$\large\boxed{\begin{array}{l}\rm f(x)=\sqrt{\dfrac{x-2}{1-x}}\\\rm \dfrac{x-2}{1-x}\geqslant0\\\underline{\sf fac_{\!\!,}a}\\\rm g(x)=x-2~e~h(x)=1-x\\\rm deseja-se\,saber\,para\,quais\,valores\,de\,x\\\rm teremos\,\dfrac{g(x)}{h(x)}\geqslant0.\\\rm\,para\,isso\,fac_{\!\!,}amos\,o\,estudo\,do\,sinal\\\rm para\,cada\,func_{\!\!,}\tilde ao\,separadamente\\\rm por\,fim\,mostra-se\,o\,quadro\,de\,sinais\\\rm e\,d\acute a-se\,a\,resposta\,conveniente.\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\sf zero\,de\,g(x):}\\\rm x-2=0\\\rm x=2\\\underline{\sf Estudo\,do\,sinal\,de\,g(x)}\\\rm g(x)>0\iff x>2\\\rm g(x)<0\iff x<2\\\rm \\\underline{\sf zero\,de\,h(x):}\\\rm 1-x=0\\\rm x=1\\\underline{\sf estudo\,do\,sinal\,de\,h(x)}\\\rm h(x)>0\implies x<1\\\rm h(x)<0\implies x>1\\\underline{\sf Observe\,o\,anexo.}\\\rm montando\,o\,quadro\,sinal\\\rm e\,assinalando\,a\,resposta\\\rm temos\,que\\\rm Dom\,f(x)=\{x\in\mathbb{R}/1<x\leqslant2\}\end{array}}$