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I've been trying to come up with a one-variable function that roughly matches this graph:

enter image description here

The closest I've gotten is $ f(d) = \dfrac{\log(c \cdot (1 - \frac{d}{c}))}{\log c} $, but this approaches $y=0$ at $x=c-1$, and steadily declines from $x=0$, instead of sharply.

Are there any functions or tricks I could look into to develop this further?

Thank you.

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    Have you tried piecewise-defined functions? This looks like you could approximate it well with a parabola near 0 (i.e., $1 - x^2$), a square root near $c$ (i.e., $\sqrt{c-x}$) and some thing like a third-degree polynomial or a rational function inbetween.2012-05-06
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    Something like $\exp(-x^2)-1/(5-x) +1$ comes close. The exponential term gives you the "left part" of the graph (with the inflection point) and the $1/(5-x)$ term gives you the "right part". Now you can scale (replace $x$ with $ax$ to scale horizontally; multiply the whole thing by $a$ to scale vertically) this to obtain what you want.2012-05-06
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    That's beautiful, @DavidMitra! Thank you very much, and for the explanation.2012-05-06
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    You could also try doing something in polar coordinates.2012-06-07

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