1. If A ⊆R that is A is a proper subset of R then f: A→B is called a real variable function.
2. If B ⊆R or B is a proper subset of R then f: A→B is called a real valued function.
3. If both A ⊆R and B ⊆R then the function f: A→B is called a real function.

Consider the following functions.

Polynomial function

A function of the form a₀xⁿ + a₁ x^n-1 + a₂x^n-2 + ..... an where a₀≠ 0, a₁, a₂ .... a_n are real constants and n is a positive integer or zero, is called a polynomial function of degree n. We generally write this as

f(x) = a₀xⁿ + a₁ x^n-1 + ..... a_n

Examples:

             x² + 2x + 3

             3x⁵ - 5x⁴ + 7x³ - 6x² - 5x -1

Rational function

A function of the form p(x) / q(x) where p(x) and q(x) are polynomial functions and q(x) ≠0 is called a rational function.

Its domain = R - {x|q(x)=0}

Example:

is a rational function such that 6x² - 3x + 5 ≠0.

Even and odd functions

If f(x) is a function such that f(-x) = f(x) for every x in its domain, then f is called an even function.

Examples:

             x², x⁴, x²-1 and |x| are all even function since

        If f(x) = x²

         f(-x) = (-x)²

                 = x² = f(x)

If f(x) is a function such that f(-x) = -f(x) for every x in its domain then f is called an odd function.

Examples:

        x, x³, 3x³-x are odd.

        Let f(x) = 5x³-x

             f(-x) = 5(-x)³ - (-x)

                    = -5x³ + x

                    = -(5x³-x)

                    = -f(x)

Exponential function

A function f(x) = a^x a>0 is called an exponential function.

Domain of f is R

Range = { x | 0 ≤ x < ∞}

Example

f: R→R+ such that f(x) = 2^x is a bijection.

R+ is the set of all positive real numbers.

Logarithmic function

A function f(x) = loga^x a ≠1 where a and x are positive numbers is called a logarithmic function.

Domain of f = { x | 0 < x < ∞}

Range of f = { x | - ∞ < x < ∞}

Example

f :R+→R such that f(x) = log2^x, this function is called a bijection.

Also, f-1: R→R+ such that

f-1(x) = 2x.

Try these questions:

1. Name the type of function.
   1. Name the type of function.
   2. f(x) = 3x³- 4x² + 7x – 1
   3. g(x) = 5x
   4. h(y) = log 1/3 y
   5. 
   
   
   1. Polynomial function
   2. Exponential function
   3. Logarithmic function
   4. Rational function