Algebraic Functions

 (1) Polynomial Function

A function say P of the form

P(x) = a_n xⁿ+a_n-1 x^n-1+a_n-2 x^n-2+a_n-3 x^n-3+. . . + a₂ x²+ a₁ x+a₀

for all where the coefficient a_n, a_n-1,  a_n-2, . . . , a₂, a₁, a₀ are real numbers and the exponents are non-negative integers, is called a polynomial function.

        The domain and range of P(x) are in general subsets of real numbers. as If a≠0, then P(x) is called a polynomial function of degree n and a, is the leading coefficient of Px).

For éxample, P(x)=2x⁴-3x³ +2x-1 this is a polynomial function of degree 4 with leading cofficient 2.

(2) Linear Function

        If the degree of a polynomial function is 1, then it is called a linear function. A linear function is of the form f(x) = ax +b(a≠0)  where a, b are real numbers. For example f(x)=3x+4,or y=3x +4 is a linear function. Its domain and range are the set of real numbers

(3) Identity Function

For any set X, a function I:X→X of the form I(x)=x for all x ∈ X, is called an identity function. Its domain and range is the set X itself.

In particular if X=R. then I(x) = x, for all x ∈ R. is the identity function.

 (4) Constant function

Let A and S be the sets of real numbers. A function denoted as C  : A→S  defined by C(x) = a for all x ∈ X, a ∈ Y  and fixed is called a constant function. For example, C:R→R defined by C(x) = 2. for all x ∈ R is a constant function.

 (5) Rational function

A function R(x) of the form  where P(x) and Q(x) are the polynomials functions and Q(x) ≠ 0 is called rational function. The domain and range is the R(x) is the set of all real numbers x for which  Q(x) ≠ 0.