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## 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.
 a. f(x) = 3x3- 4x2 + 7x – 1 b. g(x) = 5x c. h(y) = log 1/3 y d.
1. Polynomial function

2. Exponential function

3. Logarithmic function

4. Rational function