## 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 xn-1 + a2xn-2 + ..... an where a0≠ 0, a1, a2 .... an 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 xn-1 + ..... an

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) = ax a>0 is called an exponential function.

Domain of f is R

Range = { x | 0 ≤ x < ∞}

Example

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

R+ is the set of all positive real numbers.

#### Logarithmic function

A function f(x) = logax 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) = log2x, 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