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I've got this function: $\frac{x^2}{x^2+(1-x)^2}$ ; it gives me this blue graph (in zero - one range): enter image description here

Could you help me find function to achieve graph close to red one?

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What formula did you use to generate the red graph? That might give some idea of how to modify the function of the blue graph. –  coffeemath Feb 4 '13 at 9:30
    
I used Fireworks to add it. I need formula for red one. I thought it would be easier to modify what I already have but it can be completely different function. –  Martin Feb 4 '13 at 9:43

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Try replacing $x$ everywhere by $\sqrt{x}$ (or other power of $x$ less than the first power). When I graphed your function $f(x)$ along with $$g(x)=\frac{x}{x+(1-\sqrt{x})^2}$$ on the interval $[0,1]$ the $f(x)$ graph looked like your blue graph, while the $g(x)$ graph looked much like your red graph.

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Your function improves end part but the beginning doesn't change at all. It doesn't have to be based on my function. –  Martin Feb 4 '13 at 10:08
    
On my screen the $g$ function goes up first at the left end $0$, way before the $f$ goes up. –  coffeemath Feb 4 '13 at 10:46

Although I am not quite sure of what you are looking for, you may try the general form

$$g(x) = \tanh^n{(a x)}$$

Here are some plots for $n=3$, $n \in \{2,3,\ldots,10\}$:

enter image description here

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