Editing Algebraic function (section)

=== 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 (mathematics)|division]], and taking an [[nth root|''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]] <math>y = p(x)</math> is an algebraic function, since it is simply the solution ''y'' to the equation

:<math> y-p(x) = 0.\,</math>

More generally, any [[rational function]] <math>y=\frac{p(x)}{q(x)}</math> is algebraic, being the solution to

:<math>q(x)y-p(x)=0.</math>

Moreover, the ''n''th root of any polynomial <math display="inline">y=\sqrt[n]{p(x)}</math> is an algebraic function, solving the equation

:<math>y^n-p(x)=0.</math>

Surprisingly, the [[inverse function]] of an algebraic function is an algebraic function.  For supposing that ''y'' is a solution to

:<math>a_n(x)y^n+\cdots+a_0(x)=0,</math>

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,

:<math>b_m(y)x^m+b_{m-1}(y)x^{m-1}+\cdots+b_0(y)=0.</math>

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''&thinsp;=&thinsp;''x''<sup>2</sup> fails the [[horizontal line test]]: it fails to be [[one-to-one function|one-to-one]].  The inverse is the algebraic "function" <math>x = \pm\sqrt{y}</math>. 
Another way to understand this, is that the [[Set (mathematics)|set]] of branches of the polynomial equation defining our algebraic function is the graph of an [[algebraic curve]].