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Introduction and overview

The informal definition of an algebraic 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 n th 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 y = p (x) {\displaystyle y=p(x)} is an algebraic function, since it is simply the solution y to the equation

More generally, any rational function y = p (x) q (x) {\displaystyle y={\frac {p(x)}{q(x)}}} is algebraic, being the solution to

Moreover, the n th root of any polynomial y = p (x) n {\textstyle y={\sqrt{p(x)}}} 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 = x fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" x = ± y {\displaystyle x=\pm {\sqrt {y}}}. 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.