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Cuzkatzimhut (talk | contribs) No edit summary |
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\exp (f ( \ln (x))) & \mbox{if } x \in (1,\infty), \\
\ln (f ( \exp (x))) & \mbox{if } x \in (-\infty,0). \\
\end{cases}
</math>
[[File:Half-exponential_function.png|thumb|right|300px|Example of a half-exponential function]]
A simple example, which leads to ''ƒ'' having a continuous first derivative everywhere, is to take <math>A=\tfrac12</math> and <math>g(x)=x+\tfrac12</math>, giving
:<math> f (x) =
\begin{cases}
\log_e\left(e^x +\tfrac12\right) & \mbox{if } x \le -\log_e 2, \\
e^x - \tfrac12 & \mbox{if } -\log_e 2 \le x \le 0, \\
x +\tfrac12 & \mbox{if } 0 \le x \le \tfrac12, \\
e^{x-1/2} & \mbox{if } \tfrac12 \le x \le 1 , \\
x \sqrt{e} & \mbox{if } 1 \le x \le \sqrt{e} , \\
e^{x / \sqrt{e}} & \mbox{if } \sqrt{e} \le x \le e , \\
x^{\sqrt{e}} & \mbox{if } e \le x \le e^{\sqrt{e}} , \\
e^{x^{1/\sqrt{e}}} & \mbox{if } e^{\sqrt{e}} \le x \le e^e , \ldots\\
\end{cases}
</math>
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