How to invert this function? $ y = e^{\arctan(x^5)} $
How to invert this function? (Inverse exponential function with arctan)
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$\begingroup$
functions
inverse
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0A point of grammar: "inverse" is a noun; "invert" is a verb. The English language is rather chaotic about things like this. – 2012-10-25
3 Answers
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$\newcommand{\leftlong}{\longleftarrow\!\shortmid}$ What gets done last gets undone first: $ \begin{array}{rcccccl} x & \longmapsto & x^5 & \longmapsto & \arctan(x^5) & \longmapsto & \exp(\arctan(x^5)) = y \\[12pt] \sqrt[5]{\tan(\log_e y)} & \leftlong & \tan(\log_e y) & \leftlong & \log_e y & \leftlong & y \end{array} $
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I guess by "solve", you mean "find the inverse $x=f(y)$".
$y=e^{\arctan(x^{5})}\Leftrightarrow \log{y}=\arctan(x^{5})\Leftrightarrow \tan{(\log{y})}=x^{5}\Leftrightarrow(\tan(\log{y}))^{1/5}=x.$
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0@Lord_Farin The last biconditional holds perfectly well (there is always a fifth root of $x \in \mathbb{R}$), and it's reasonable to assume that the student understands that $\tan(x)$ has to be restricted to a suitable interval. (**Edit:** Whoops, I thought this was recent.) – 2012-11-01
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${}x=\sqrt[5]{\tan(\log y)}$