Will operations on a purely random function also be random?

Question

Let's assume that a function random(x,y) gives a completely random number as output which will be in between x and y (not a pseudo random function) such that function's output cannot be statistically predetermined then will a function like f(x,y)=random(x,y)+1000 or g(x,y)=random(x,y)*21 will also be a purely random function producing totally random output. If yes then why and if no then why ?

Answer 1

The values of random will be uniformly distributed in the interval $(x,y)$. I think that's what you mean by "completely random". But the values of your function need not be. In your squaring example, suppose $(x,y) = (0,1)$. If you look at the graph of the squaring function you'll see that random$(0,1)^2$ is more likely to be near $0$ than near $1$.

The transformed distribution will be uniform when the transformation is linear, like the $f$ in your question.

Answer 2

It is true that if $X\sim\operatorname{Unif}(a,b)$, then $\alpha X+\beta\sim \operatorname{Unif}(\beta+\alpha a,\beta+\alpha b)$ if $\alpha>0$ and $\alpha X+\beta\sim \operatorname{Unif}(\beta+\alpha b,\beta+\alpha a)$ if $\alpha<0$. So, all approximations of this phenomenon should do it as well.

All notable distributions behave well with translation and rescaling, where "well" indicates a behaviour of the relevant coefficients which should strike the knowledgeable person as intuitive.