I will show you how to do this and you can modify it for other cases. Your function goes from -1 to 1 which is 2 units. Let's multiply it by $\frac 1 2$ so it goes from $-\frac 1 2$ to $\frac 1 2$ and move it up $\frac 1 2$. So now it looks like$$

f(x) = \frac 1 2\frac {x}{|x|+1}+\frac 1 2$$This will have asymptotes $y=0$ and $y=1$ You want the upper asymptote to be $.95$ so let's multiply it by $.95$. So now$$

f(x) = .95\left (\frac 1 2\frac {x}{|x|+1}+\frac 1 2 \right)$$

So far this graph has the right shape and desired asymptotes. Now this function takes the value $.05$ way to the left of $x=0$. If you set $f(x) = .05$ and solve it you get $x = -8.5$. So $f(-8.5)=.05$ but you want $f(0) = .05$. So now replace $x$ by $x - 8.5$ in your equation. Now your equation is$$

f(x)= .95\left (\frac 1 2\frac {x-8.5}{|x-8.5|+1}+\frac 1 2 \right)$$

Translating it horizontally didn't change the horizontal asymptotes and now $f(0) = .05$ All that is left is to get $f(1)=.75$. Currently the value of $x$ that gives $f(x) = .75$ is $x=9.875$. So if you scale the $x$ axis by replacing $x$ by $9.875x$, that should do it. Your final formula becomes$$

f(x)=.95\left (\frac 1 2\frac {9.875x-8.5}{|9.875x-8.5|+1}+\frac 1 2 \right)$$

Here's a graph of the final result (click to enlarge):