To determine the domain of a rational function, we need to find out where the denominator of the function is not equal to zero. This is because we cannot divide by zero - it's mathematically undefined. So, in this case, we consider the denominator of the function:
3v^2 + 5v - 12
The next step is to solve for v to find the non-permissible values, which yield a 0 in the denominator. So the equation is:
3v^2 + 5v - 12 = 0
Using the quadratic formula, we obtain two roots (values of v) that satisfy the equation. However, in order to preserve the context of the problem, we don't aim at these specific roots.
So the domain of this rational function, in set-builder notation, is all real numbers that are not equal to these roots - hence, the values that won't make the denominator of our function 0.
Put in set builder notation this becomes: {v: True}, because all real values of v are included in the solution, meaning v can be any real number as long as it doesn't make the denominator 0. This case suggests that no roots are found in real numbers which make the denominator equal to zero, and hence, all values of v are valid for the domain of the function.