What simple functions return equally distributed random values in an arbitrary given range? For programming purposes I want a function f(x,R) that given a certain seed x returns the same random value every time, in an arbitrary range R. But, I also want the output to be equally distributed. Is this mathematically possible?
For example, I can take the 1st - nth decimal places of sin(x) such that 10^n > R and divide by R, but I'm sure that this isn't evenly distributed even for R = 10.
Otherwise (veering slightly into Stack Overflow territory here) are there existing well-established functions/algorithms for producing (pseudo)random numbers in this fashion?
 A: To give a StackOverflowy answer, how about generating a random value in your given range for every new input and return the previously generated value if the input appears again? You'd have to store every pair, though.
I think this is the only way to introduce randomness into your setup. 
The example given in your question is not random at all, it is deterministic. Are you sure that you are really looking for randomness instead of maybe a hash function?
A: As per your comment, it seems that you actually want some deterministic function $f(N,x)$ that returns a integer in $[1..N]$, and which, for a fixed $N$ and for different values of $x$ returns values that appear random in that interval. Still, this requirement is not totally clear, because one should stipulate a distribution of $x$. But, if I understand your scenario, one can assume that $x$ is uniform in some large interval (say, the range the full range of integers in you platform-language). So, a simple recipe would be to use a standard pseudorandom generator and use $x$ as a seed. For example, in Java:
static  int getrand(int N,long x) {
    Random r = new Random(x);  // random number generator, with x as seed
    return r.nextInt(N)+1;     // pseudoradom number in 1..N range 
} 

Or you can directly implement your own pseudorandom routine, eg,
static  int getrand2(int N,long x) {
    long a=1664525;  
    long c=1013904223;
    long res = (a*x+c)&0xFFFFFFFF; 
    return (int)((res % N)+1);
}

