exponential function with values between 0 and 1 for x values between 0 and 1

Question

I am looking for a function that fits well to be used as a weight with exponential behavior. My x values are between 0 and 1, and i would like this function to behave exponentially, so that only x values close to 1 have y values close to 1 (and vice versa x values close to 0 have y values close to 0).

The function should behave asymptotically, so that:

$\lim_{x \rightarrow 1} f(x) = 1$ and $\lim_{x \rightarrow 0} f(x) = 0$

Answer 1

$$y=\frac{a^x-1}{a-1}, a>1$$

Will satisfy the conditions you've mentioned.

Answer 2

$$f(x)=e^{1-\frac{1}{x^2}}$$should be ok.