What is Algebraic Function?

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.

Examples of such functions are:

  • f(x)=1/x
  • f(x)={\sqrt {x}}
  • {\displaystyle f(x)={\frac {\sqrt {1+x^{3}}}{x^{3/7}-{\sqrt {7}}x^{1/3}}}}

Some algebraic functions, however, cannot be expressed by such finite expressions (this is Abel–Ruffini theorem). This is the case, for example, of the Bring radical, which is the function implicitly defined by

 {\displaystyle f(x)^{5}+f(x)+x=0}.

In more precise terms, an algebraic function of degree n in one variable x is a function  y= y=f(x)hat satisfies a polynomial equation

 a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0

where the coefficients ai(x) are polynomial functions of x, with coefficients belonging to a set S. Quite often, S=\mathbb {Q} , and one then talks about “function algebraic over \mathbb {Q} “, and the evaluation at a given rational value of such an algebraic function gives an algebraic number.

A function which is not algebraic is called a transcendental function, as it is for example the case of \exp(x),\tan(x),\ln(x),\Gamma (x). A composition of transcendental functions can give an algebraic function-f(x)=\cos(\arcsin(x))={\sqrt {1-x^{2}}}.

As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, but n functions, sometimes also called branches. Consider for example the equation of the unit circle: y^{2}+x^{2}=1.\, This determines y, except only up to an overall sign; accordingly, it has two branches: y=\pm {\sqrt {1-x^{2}}}.\,

An algebraic function in m variables is similarly defined as a function y which solves a polynomial equation in m + 1 variables:

p(y,x_{1},x_{2},\dots ,x_{m})=0.\,

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