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Question Number 73649 by aliesam last updated on 14/Nov/19

$$findtherange$$$$$$$$f\left(x\right)=\frac{2}{6-\sqrt{x+2}}$$

Commented by mathmax by abdo last updated on 14/Nov/19

$$x\in D_f\iff x⩾-2and\sqrt{x+2}\ne 6\iff x⩾-2andx+2\ne 36\iff x⩾-2$$$$andx\ne 34\Rightarrow D_f=[-2,34[\cup ]34,+\mathrm{\infty }[$$$$f^{{}^{\prime }}\left(x\right)=2(-\frac{-\frac{1}{2\sqrt{x+2}}}{{(6-\sqrt{x+2})}^2})=\frac{1}{\sqrt{x+2}{(6-\sqrt{x+2})}^2}>0\Rightarrow fisincreazing$$$$on{D}_{f}wehavef(-2)=\frac{1}{3}and{lim}_{x\to {34}^{-}}f\left(x\right)=+\mathrm{\infty }$$$${lim}_{x\to {34}^{+}}f\left(x\right)=-\mathrm{\infty }\Rightarrow f([-2,34\left[\right)=[\frac{1}{3},+\mathrm{\infty }[and$$$$f\left(\right]34,+\mathrm{\infty }\left[\right)=]-\mathrm{\infty },0[variationoff$$$$x-234+\mathrm{\infty }$$$$f^{{}^{\prime }}+\mid \mid +$$$$f\frac{1}{3}inc+\mathrm{\infty }\mid \mid -\mathrm{\infty }inc0$$

Commented by malwaan last updated on 14/Nov/19

$$\mathit{x}⩾-2\mathit{a}\mathit{n}\mathit{d}\mathit{x}\ne 34$$$$\Rightarrow -2⩽\mathit{x}<34\vee \mathit{x}>34$$$$-2⩽\mathit{x}<34\Rightarrow 0⩽\mathit{x}+2<36$$$$\Rightarrow 0⩽\sqrt{\mathit{x}+2}<6\Rightarrow 0⩾-\sqrt{\mathit{x}+2}>-6$$$$\Rightarrow 6⩾6-\sqrt{\mathit{x}+2}>0$$$$\Rightarrow \frac{1}{3}⩽\frac{2}{6-\sqrt{\mathit{x}+2}}<\mathrm{\infty }$$$$\Rightarrow \frac{1}{3}⩽\mathit{y}<\mathrm{\infty }....\left(1\right)$$$$\mathit{x}>34\Rightarrow 34<\mathit{x}<\mathrm{\infty }$$$$\Rightarrow 36<\mathit{x}+2<\mathrm{\infty }$$$$\Rightarrow 6<\sqrt{\mathit{x}+2}<\mathrm{\infty }$$$$\Rightarrow -6>-\sqrt{\mathit{x}+2}>-\mathrm{\infty }$$$$\Rightarrow 0>6-\sqrt{\mathit{x}+2}>-\mathrm{\infty }$$$$\Rightarrow -\mathrm{\infty }<\frac{2}{6-\sqrt{\mathit{x}+2}}<0$$$$\Rightarrow -\mathrm{\infty }<\mathit{y}<0....\left(2\right)$$$$\mathit{f}\mathit{r}\mathit{o}\mathit{m}\left(1\right);\left(2\right)$$$$\Rightarrow \mathit{t}\mathit{h}\mathit{e}\mathit{r}\mathit{a}\mathit{n}\mathit{g}\mathit{e}=$$$$[1╱3;\mathrm{\infty }[\cup ]-\mathrm{\infty };0[$$

Answered by MJS last updated on 14/Nov/19

$$\mathrm{defined}\mathrm{for}x⩾-2\wedge x\ne 34$$$$f(-2)=\frac{1}{3}$$$$\underset{x\to {34}^{-}}{\lim }f\left(x\right)=+\mathrm{\infty }$$$$\underset{x\to {34}^{+}}{\lim }f\left(x\right)=-\mathrm{\infty }$$$$\underset{x\to +\mathrm{\infty }}{\lim }f\left(x\right)=0$$$$\Rightarrow -\mathrm{\infty }<y⩽0\vee \frac{1}{3}⩽y<+\mathrm{\infty }$$

Commented by aliesam last updated on 14/Nov/19

$$godblessyousir$$

Commented by aliesam last updated on 14/Nov/19

$$butithinkthat$$$$$$$$-\mathrm{\infty }<y<0$$

Commented by MJS last updated on 14/Nov/19

$$\mathrm{yes}\mathrm{you}\mathrm{are}\mathrm{right},{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{a}\mathrm{typo}$$

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