function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. This is something of an oversimplification; because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function is an algebraic function, since it is simply the solution y to the equation More generally, any rational function is algebraic, being the solution to Moreover, the nth root of any polynomial is an algebraic function, solving the equation Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution to for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms, Writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" . Another way to understand this, is that the set of branches of the polynomial equation defining our algebraic function is the graph of an algebraic curve. The role of complex numbers From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in y) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized. Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express the function in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (see casus irreducibilis). For example, consider the algebraic function determined by the equation Using the cubic formula, we get For the square root is real and the cubic root is thus well defined, providing the unique