The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function; the set of all real numbers for which a function is mathematically defined.

2x^2 + 6x - 8 = 0
2(x - 1)(x + 4) = 0
x - 1 = 0 or x + 4 = 0
x = 1 or x = - 4

2 + 4x - 45 = 0
(4x) + (2 - 45) = 0
4x - 43 = 0
4x = 0 + 43
4x = 43
x = 43/4

Purpose: the reason for which something exists or is done, made, used, etc.