Help me to find function to this graph

Question

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

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

Answer 1

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.

Answer 2

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 $n=3$, $n \in \{2,3,\ldots,10\}$: