Đáp án:
$a)-\dfrac{1}{2} \le m \le 1\\ b)m \in \varnothing\\ c) m <-2\\ d) \left[\begin{array}{l} m = 0\\ m<-5\end{array} \right.$
Giải thích các bước giải:
$y=\sqrt{x-m}+\dfrac{1}{x+2m}\\ =\sqrt{x-m}+\dfrac{1}{x-(-2m)}$
$a)$Hàm số xác định trên $(1;+\infty)$
$\Rightarrow \left\{\begin{array}{l} x-m\ge 0 \ \forall x \in (1;+\infty)\\ x-(-2m) \ne 0\ \forall x \in (1;+\infty)\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m \le x \ \forall x \in (1;+\infty)\\ (-2m) \ne x\ \forall x \in (1;+\infty)\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m \le 1\\ -2m \le 1\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m\le 1\\ m \ge -\dfrac{1}{2}\end{array} \right.\\ \Leftrightarrow -\dfrac{1}{2} \le m \le 1$
$b)$Hàm số xác định trên $[-2;+\infty)$
$\Rightarrow \left\{\begin{array}{l} x-m \ge 0 \ \forall x \in [-2;+\infty)\\ x-(-2m) \ne 0\ \forall x \in [-2;+\infty)\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m \le x \ \forall x \in [-2;+\infty)\\ (-2m) \ne x\ \forall x \in [-2;+\infty)\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m\le -2\\ -2m < -2\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m \le -2\\ m >1\end{array} \right.\\ \Leftrightarrow m \in \varnothing$
$c)$Hàm số xác định trên $[-1;4]$
$\Rightarrow \left\{\begin{array}{l} x-m \ge 0 \ \forall x \in [-1;4]\\ x-(-2m) \ne 0\ \forall x \in [-1;4]\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m\le x \ \forall x \in [-1;4]\\ (-2m) \ne x\ \forall x \in [-1;4]\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m \le -1\\ \left[\begin{array}{l} -2m<-1 \\ -2m>4\end{array} \right.\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m \le -1\\ \left[\begin{array}{l} m>\dfrac{1}{2} \\ m<-2\end{array} \right.\end{array} \right.\\ \Leftrightarrow m <-2$
$d)$Hàm số xác định trên $(0;10]$
$\Rightarrow \left\{\begin{array}{l} x-m \ge 0 \ \forall x \in (0;10]\\ x-(-2m) \ne 0\ \forall x \in (0;10]\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m\le x \ \forall x \in (0;10]\\ (-2m) \ne x\ \forall x \in (0;10]\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m\le 0\\ \left[\begin{array}{l} -2m\le 0 \\ -2m>10\end{array} \right.\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} m\le 0\\ \left[\begin{array}{l} m \ge 0\\ m<-5\end{array} \right.\end{array} \right.\\ \Leftrightarrow \left[\begin{array}{l} m = 0\\ m<-5\end{array} \right.$
