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Question Number 27608 by chernoaguero@gmail.com last updated on 10/Jan/18

$$\mathrm{f}\left(\mathrm{x}\right)=\ln (\mathrm{x}+\sqrt{{\mathrm{x}}^2+1})$$$$$$$$\mathrm{find}{\mathrm{f}}^{-1}\left(\mathrm{x}\right)$$$$\mathrm{plzz}\mathrm{help}$$

Commented by abdo imad last updated on 11/Jan/18

$$f\left(x\right)=y\iff f^{-1}\left(y\right)=xbutf\left(x\right)=y\iff ln(x+\sqrt{x^2+1})=y$$$$x+\sqrt{x^2+1}=e^y\iff \sqrt{x^2+1}=e^y-xand{e}^y>x$$$$\iff x^2+1={(e^y-x)}^2\iff x^2={(e^y-x)}^2-1$$$$\iff x^2=e^{2y}-2e^{y}x+x^2-1\iff 2e^{y}x=e^{2y}-1$$$$\iff x=\frac{e^{2y}-1}{2e^y}=\frac{e^y-e^{-y}}{2}=sinhy$$$$f^{-1}\left(x\right)=\frac{e^x-e^{-x}}{2}=shx.$$$$$$$$$$

Commented by chernoaguero@gmail.com last updated on 11/Jan/18

$$\mathrm{Thank}\mathrm{u}\mathrm{sir}\mathrm{but}\mathrm{what}\mathrm{do}\mathrm{u}\mathrm{mean}$$$$\mathrm{by}\mathrm{sinhy}\mathrm{an}\mathrm{shx}$$

Commented by Joel578 last updated on 11/Jan/18

$$\mathrm{sh}x\mathrm{is}\sinh x$$

Commented by abdo imad last updated on 11/Jan/18

$$sinhx=shxitsonlyanotation.$$

Commented by chernoaguero@gmail.com last updated on 11/Jan/18

$$\mathrm{ok}\mathrm{thank}$$

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