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

Consider the following functions.

A function of the form a_{0}x^{n} + a_{1} x^{n-1} + a_{2}x^{n-2} + ..... an where a_{0}≠ 0, a_{1}, a_{2} .... 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) = a_{0}x^{n} + a_{1} x^{n-1} + ..... a_{n}

**Examples**:** **

x^{2} + 2x + 3

3x^{5} - 5x^{4} + 7x^{3} - 6x^{2} - 5x -1

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 6x^{2} - 3x + 5 ≠0.

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**:

x^{2}, x^{4}, x^{2}-1 and |x| are all even function since

If f(x) = x^{2}

f(-x) = (-x)^{2}

= x^{2} = 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, x^{3}, 3x^{3}-x are odd.

Let f(x) = 5x^{3}-x

f(-x) = 5(-x)^{3} - (-x)

= -5x^{3} + x

= -(5x^{3}-x)

= -f(x)

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.

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.

- Name the type of function.
- Name the type of function.
- f(x) = 3x
^{3}- 4x+ 7x – 1^{2} - g(x) = 5x
- h(y) = log 1/3 y

- Polynomial function
- Exponential function
- Logarithmic function
- Rational function

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