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Question Number 174635 by mnjuly1970 last updated on 06/Aug/22

Answered by aleks041103 last updated on 06/Aug/22

$${tanh}^{-1}(-x)=-{tanh}^{-1}\left(x\right)$$$$\Rightarrow {tanh}^{-1}isodd.$$$$(-x)=-\left(x\right)$$$$\Rightarrow xisodd.$$$$ln(-x+\sqrt{1+{(-x)}^2})=$$$$=ln(\sqrt{1+x^2}-x)=$$$$=ln\left(\frac{(\sqrt{1+x^2}-x)(\sqrt{1+x^2}+x)}{x+\sqrt{1+x^2}}\right)=$$$$=ln((1+x^2)-x^2)-ln(x+\sqrt{1+x^2})=$$$$=-ln(x+\sqrt{1+x^2})$$$$\Rightarrow ln(x+\sqrt{1+x^2})isodd.$$$$\Rightarrow f\left(x\right)=odd.odd.odd=odd$$$$\Rightarrow f(-x)=-f\left(x\right)$$$$(f\circ f)(-x)=f\left(f(-x)\right)=f(-f\left(x\right))=-f\left(f\left(x\right)\right)$$$$\Rightarrow (f\circ f)(-x)=-(f\circ f)\left(x\right)$$$$\Rightarrow (f\circ f)isodd$$$$\Rightarrow {(f\circ f)}^{\prime }iseven$$$$\Rightarrow {(f\circ f)}^{\prime }\left(\frac{1}{\pi }\right)={(f\circ f)}^{\prime }(-\frac{1}{\pi })$$$$\Rightarrow {(f\circ f)}^{\prime }\left(\frac{1}{\pi }\right)-{(f\circ f)}^{\prime }(-\frac{1}{\pi })=0$$

Commented by mnjuly1970 last updated on 06/Aug/22

$$great...excellent$$

Answered by Mathspace last updated on 06/Aug/22

$$fest\rho aire\Rightarrow fofim\rho aireand$$$${\left(fof\right)}^{{}^{\prime }}est\rho aire\Rightarrow$$$${\left(fof\right)}^{{}^{\prime }}(-\frac{1}{\pi })={\left(fof\right)}^{{}^{\prime }}\left(\frac{1}{\pi }\right)\Rightarrow$$$${\left(fof\right)}^{{}^{\prime }}\left(\frac{1}{\pi }\right)-{\left(fof\right)}^{{}^{\prime }}(-\frac{1}{\pi })=0$$

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