$a)$  

$f\left(x\right)=\frac{x}{\left|x\right|-5}$  

$D:$  

$\left|x\right|-5\ne 0$  

$\left|x\right|\ne 5$  

$x\ne 5\wedge x\ne -5$  

$D=\mathbb{R}\backslash \left\{-5;5\right\}$  

 

$f\left(x\right)=\frac{x}{\left|x\right|-5}=0$  

$\underline{x=0}$  

 

$b)$  

$f\left(x\right)=\frac{x+2}{\left|x\right|+4}$  

$D:$  

$\left|x\right|+4\ne 0$  

$\left|x\right|\ne -4$  

Modół z dowolnej liczby jest zawsze nieujemny.

$D=\mathbb{R}$  

 

$f\left(x\right)=\frac{x+2}{\left|x\right|+4}=0$  

$x+2=0$  

$\underline{x=-2}$    

 

$c)$  

$f\left(x\right)=\frac{\left|x-1\right|}{\left|x\right|-1}$  

$D:$  

$\left|x\right|-1\ne 0$  

$\left|x\right|\ne 1$  

$x\ne 1\wedge x\ne -1$  

$D=\mathbb{R}\backslash \left\{-1;1\right\}$  

 

$f\left(x\right)=\frac{\left|x-1\right|}{\left|x\right|-1}=0$  

$\left|x-1\right|=0$  

$x=1\notin D$  

Brak miejsc zerowych.

 

$d)$  

$f\left(x\right)=\frac{3-\left|x\right|}{x\left(2x-6\right)}$  

$D:$  

$x\left(2x-6\right)\ne 0$  

$x\ne 0\wedge 2x-6\ne 0$  

$x\ne 0\wedge x\ne 3$  

$D=\mathbb{R}\backslash \left\{0;3\right\}$      

 

$f\left(x\right)=\frac{3-\left|x\right|}{x\left(2x-6\right)}=0$  

$3-\left|x\right|=0$  

$\left|x\right|=3$  

$x=3\notin D\vee x=-3$  

$\underline{x=-3}$  

 

$e)$  

$f\left(x\right)=\frac{x\left(x+1\right)}{1-\left|x+1\right|}$  

$D:$  

$1-\left|x+1\right|\ne 0$  

$\left|x+1\right|\ne 1$  

$x+1\ne 1\wedge x+1\ne -1$  

$x\ne 0\wedge x\ne -2$  

$D=\mathbb{R}\backslash \left\{-2;0\right\}$  

 

$f\left(x\right)=\frac{x\left(x+1\right)}{1-\left|x+1\right|}=0$  

$x\left(x+1\right)=0$  

$x=0\notin Dx=-1$  

$\underline{x=-1}$  

 

$f)$   

$f\left(x\right)=\frac{x^2-9}{1-\left|x+2\right|}$  

$D:$  

$1-\left|x+2\right|\ne 0$  

$\left|x+2\right|\ne 1$  

$x+2\ne 1\wedge x+2\ne -1$  

$x\ne -1\wedge x\ne -3$  

$D=\mathbb{R}\backslash \left\{-3;-1\right\}$  

 

$f\left(x\right)=\frac{x^2-9}{1-\left|x+2\right|}=0$  

$x^2-9=0$  

$x=3\vee x=-3\notin D$  

$\underline{x=3}$