More Functions

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

1. Name the type of function.
   

1. Polynomial function
   
   
2. Exponential function
   
   
3. Logarithmic function
   
   
4. Rational function

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