A **function** is a mapping between two sets, called the **domain** and the **range**, where for every value in the domain there is a unique value in the range assigned by the function. Generally functions are defined by some formula; for example *f*(*x*) = *x*^{2}* *is the function that maps values of *x* into their square. The mapping is between all numbers (the domain) and non-negative numbers (the range).

A function is **polynomial** if it has the form

for some non-negative integer *n* (called the **degree** of the polynomial) and some constants *a*_{0}*, a*_{1}*, …, a _{n} *where

*a*≠ 0 (unless

_{n}*n*= 0). The function is

**linear**if

*n*= 1 and

**quadratic**if

*n*= 2.

We often use **limit** notation such as

as a shorthand for as *x* gets larger the value of the function *f*(*x*) **approaches** the value *a*. Some examples for various values of *f*(*x*) are as follows:

We can also evaluate the limit of a function when *x* approaches some other value. For example in the following example as *x* gets close to 0 the value of the function approaches 1.

We can also use the limit notation with series. If *x _{n}* = 1 – 1/

*n*where

*n*is a positive integer then

since the series , , , , … **converges** to 1.

Often we use a **graph** to show the relationship between *x* and *f*(*x*). A graph consists of all the points (*x,*y) in what is called the **xy plane** where *x* is in the domain of *f* and y = *f*(*x*). The graph is drawn in a rectangular grid defined by the **x-axis** (drawn horizontally) and the **y-axis** (drawn vertically) meeting at a point (0, 0) call the **origin**. In particular, the graph of y = *f*(*x*) for *f*(*x*) = *x ^{2} *is given in Figure 1.

**Figure 1 – Graph of ****y = f(x) where f(x) = x^{2}**

A **line** is a set of points (*x*, y) such that y* = bx + a* for some constants *a* and *b*. *b* is called the **slope** of the line and *a* is called the **y****-intercept**.