Functions can be broadly categorized into two groups. 

(a) Algebraic Functions 

(b) Transcendental Functions 

 Algebraic Functions: 

A function that consists of a finite number of terms involving powers and roots of the independent variable x and the four fundamental operations of addition, subtraction, multiplication, and division is called an algebraic function.

e.g. 

5x³ – 7x² + 6, \(4\sqrt{7x-12}\) , \(2x^\frac{1}{2} \) +4x-7 , \(\frac{2x+1}{2x+3}\), etc.

The particular cases of algebraic functions are:

(i) Polynomial Functions: A function f (x) = a₀ + a₁x + a₂x² + ..... + a_nxⁿ, where n ∈ N and a₁, a₂, a₃, ..... , a_n ∈ R is called a polynomial function. 

Its 

Domain is the set of real numbers 

Range is the set of real numbers.

(ii) Rational Function:  A function f : A → R, f (x) = \(\frac{P(x)}{Q(x)}\) , where Q(x) ≠ 0 is called a rational function. Here A = {x : x ∈ R} such that Q(x) ≠ 0}, and P(x) and Q(x) are polynomial functions of x. 

Ex. 

(i) \(\frac{x^2+5x+6}{x^2-3x+2}\) is a polynomial function with domain = R – {1, 2}

(\(\because\) The roots of x2 – 3x + 2 are x = 1, x = 2)

(ii) f (x) = \(\frac{1}{x}\) : Domain = R – {0}, Range = R – {0}

(iii) f (x) = \(\frac{1}{x^2}\) : Domain = R – {0}, Range = Set of positive real numbers.

(iv) f (x) = \(\frac{1}{x^3}\) : Domain = R – {0}, Range = R – {0}.

(iii) Irrational Functions: 

Functions as \(\sqrt{3x^2-7x+4}\) , \(4x^2+\sqrt{3x}\) , \(\frac{1}{\sqrt[3]{5+3x}}\)

 i.e., involving radicals or non-integral powers of x are called irrational functions.