transfer function over a uniform distribution to generate a two-fold variation, but result is also s uniform distribution

Question

I think, I am stuck at a transformation problem where I believe there is a solution, but I don't know what it is. I have uniform distribution $u_1(x) = \frac{1}{b - a}$, where $b=-1$ and $a=1$, and want to utilize it to generate two fold variation around a scalar quantity $p$. Currently, I am generating this two fold variation by using $p.(2^x)$, which makes $2^x$ the transfer function. Eventually, it gives me an exponential distribution $f(x)=p.2^{x}$. However, I want also the resulting distribution to be an uniform distribution and not exponential. Is there any such transfer function (presumably some hyperbolic function) which I can utilize over the uniform distribution $u_1(x)$ to generate a two fold variation as well as the resulting distribution is also an uniform distribution $u_2(x)$.

Answer 1

Let me rephrase the question. The problem is basically the random variable transformation. $Y=\phi(X)$ The random variable $X$ has a uniform distribution $U(-1,1)$ and I want to transform it into $Y$ which is also an uniform distribution $U(\frac{s}{2},2s)$, where $s$ is some scalar quantity. Does anyone know a transformation function $\phi$ which transforms $X$ into $Y$ with these constraints? Presently, I am using the function $Y=s2^X$, which unfortunately gives me an exponential distribution of $Y$ and not uniform distribution. Thank you in advance