Is there a way to put restrictions on the starting point for generating functions. For instance If I wanted to know how many numbers between $0$ and $50,000$ have the sum of their digits equal to $12$ or less I can easily make a generating function. However, if I wanted to see specifically how many numbers between $30,000$ and $50,000$ have the sum of their digits to $12$ or less. Is there a way cut out all the numbers before $30,000$?
My original function is $$g(x)= (1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)^5$$ then I would add up the coefficients for x raised to the $12$th, $11$th, etc and sum them.
Basically I know I want the first digit to start with a $3$ or $4$.
Note that your original function is a generating function for the digit sums of all five-digit numbers: i.e. it goes up to $99999$, not to $50000$.
If you're interested in numbers from $30000$ inclusive to $50000$ exclusive then you have a digit which is $3$ or $4$ followed by four digits which are $0$ to $9$. The generating function is just $$(x^3 + x^4)\left(\frac{x^{10}-1}{x-1}\right)^4$$
If you want to work with ranges which don't fall neatly on powers of $10$, you probably have to split into ranges which do and add or subtract. E.g. for $25000$ to $35000$ you could split into the range $25000$ to $30000$ as $(2)(5-9)(0-9)^3$ and $30000$ to $35000$ similarly.