A **function** is a rule that assigns to each input number exactly one output number. The set of input numbers to which the rule applies is called the **domain** of the function. The set of all output numbers is called the **range**.

A variable that represents *input numbers* for a function is called an **independent variable**. A variable that represents *output numbers* is called a **dependent variable**.

Output numbers such as are called **function values**; they are in the range of .

Lets be specific about the domain of a function. Unless otherwise stated, *the domain consists of all real numbers for which the equation makes sense and gives function values that are real numbers*.

**Hint,**

- We cannot divide by zero
- where does not give a real number (imaginary number)

# Types of functions

Any function in the form , where is a *constant*, is called a **constant function**.

A constant function belongs to a broader class of functions, called *polynomial functions*. in general a function of the form The number is called the **degree** of a function, and is the **leading coefficient**. Note that a nonzero constant function, such as [which can be written as ], is a polynomial function of degree 0.

A function that is the quotient of polynomial functions is called a **rational function**.

Sometimes more than one equation is needed to define a function.

This is called a **compound function**.

The function is called the **absolute function. **Recall that that the absolute value, of a real number is denoted and is defined by,

# Graphs

The **graph of a function** is simply the graph of the function . It consists of all points .

In general, *the domain consists of all x-values that are included in the graph, and the range is all y-values that are included*.

f(x) = x^2 + 2x + 3

It is clear that the domain of this function is all real numbers and the range is all reals