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mtayab1994
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Homework Statement
There were other questions before this one but i solved them all.
Find the inverse function of [tex]f(x)=arctan(\sqrt{1+x^{2}}-x)[/tex] for every x in the interval ]0,pi/2[ .That's the interval that I found when counting f(R) because f is a bijection from R to f(R). Hence f(R)=J=]0,pi/2[. ( just in case someone was wondering where i got it from?
The Attempt at a Solution
For every x in J we get :
[tex]
\begin{eqnarray*}
x=arctan(\sqrt{1+y^{2}}-y)\Rightarrow tan(x)=tan(arctan(\sqrt{1+y^{2}}-y) & \Rightarrow & tan(x)=\sqrt{1+y^{2}}-y\\
& \Rightarrow & tan(x)+y=\sqrt{1+y^{2}}\\
& \Rightarrow & tan^{2}(x)+2ytan(x)+y^{2}=1+y^{2}
\end{eqnarray*}[/tex]
then [tex]tan^{2}(x)+2ytan(x)+y^{2}=1+y^{2}\Rightarrow tan(x)[tan(x)+2y]=1\Rightarrow tan(x)+2y=\frac{1}{tan(x)}\Rightarrow tan(x)+2y=tan(\frac{\pi}{2}-x)[/tex]
then [tex]tan(x)+2y=tan(\frac{\pi}{2}-x)\Longrightarrow2y=tan(\frac{\pi}{2}-2x)\Rightarrow y=\frac{1}{2}[tan(\frac{\pi}{2}-2x)][/tex]
Is that correct, because the given answer here is: [tex]f^{-1}(x)=tan(\frac{\pi}{2}-2x)[/tex]
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