اگر $f(x)={{x}^{۲}}-۱$ و $g(x)=x+۱$ باشند، دامنه و ضابطهٔ تابع $\frac{f}{g}$ کدام است؟
1
$\left\{ \begin{align} & (\frac{f}{g})(x)=x-۱ \\ & {{D}_{\frac{f}{g}}}=R-\{۱\} \\ \end{align} \right.$
✓
✗
2
$\left\{ \begin{align} & (\frac{f}{g})(x)=x-۱ \\ & {{D}_{\frac{f}{g}}}=R-\{-۱\} \\ \end{align} \right.$
✓
✗
3
$\left\{ \begin{align} & (\frac{f}{g})(x)=\frac{{{x}^{۲}}-۱}{x+۱} \\ & {{D}_{\frac{f}{g}}}=R \\ \end{align} \right.$
✓
✗
4
$\left\{ \begin{align} & (\frac{f}{g})(x)=\frac{{{x}^{۲}}-۱}{x+۱} \\ & {{D}_{\frac{f}{g}}}=R-\{\pm ۱\} \\ \end{align} \right.$
✓
✗
$(\frac{f}{g})(x)=\frac{f(x)}{g(x)}=\frac{{{x}^{2}}-1}{x+1}=\frac{(x-1)(x+1)}{x+1}=x-1$ $({{D}_{f}}=R,{{D}_{g}}=R)\Rightarrow {{D}_{\frac{f}{g}}}={{D}_{f}}\bigcap {{D}_{g}}-\left\{ x|g(x)=0 \right\}$ $=R\bigcap R=\left\{ x|\underbrace{x+1=0}_{x=-1} \right\}=R-\left\{ -1 \right\}$