# Functions & Graphs

In this article we first give a practical definition of a **function** and then demonstrate the **graphs** of functions in the **Cartesian coordinate system** and the **polar coordinate system**.

A **function** from a set A to a set B is simply a rule which assigns to each element in the set A a unique element in the set B. The set A is called the *domain* of the function and the set B is called the *range* of the function.

When talking about a *real valued* **function** of *real numbers* (that is, when both the domain A and the range B of the **function** are the set of real numbers),
the rule of assignment is usually expressed as a formula. For example, the
**function** that assigns each real number x to *5 more than twice of* x
is expressed as

y = 2x + 5

So the function assigns 1 to 7 because

2(1)+5 = 7

Similarly

2(3)+5 = 11

2(6)+5 = 17

which says 3 and 6 are assigned to 11 and 17, respectively.

The letters such as x and y that appear in the definition of a **function** are called **variables**. The variable x is called the **independent variable** and the variable y is called the **dependent variable**. More often a more descriptive notation is used to show the dependency of the value of y on the value of x by writing

y(x) = 2x + 5

When using this notation, we usually use the letter f (or g, h, etc.) instead of y and write

f(x) = 2x + 5

With this notation we have a more descriptive way of expressing the above calculation

f(1) = 2(1)+5 = 7

f(3) = 2(3)+5 = 11

f(6) = 2(6)+5 = 17

If b = f(a), b is said to be the *value *of *f *at a. By *evaluating* the function f at a we mean finding f(a).

We may also write y = f(x) when we want to say, in general, that y is a function of x. In this case f(x) is understood to be an expression in x.

**Remark**: it is immaterial what **variables** are used for defining a **function**. That is, y = f(x) or u = f(v) or r = f(θ) can all be used to define or identify the same **function** f.

## Graphs of a Function

Just as a given point can be graphed in the Cartesian or polar coordinate systems, a given **function** f can be graphed using either of the coordinate systems because the *graph of a function* is the set of ordered pairs (a, b) where a and b are paired by the defining function f.

The **graph** of the **function** f in the **Cartesian** coordinate system (or the **Cartesian graph** of f) is the set of all points (x, y) where y = f(x), or equivalently, (x, f(x)) for x in the domain of f.

The **graph** of the **function** f in the **polar** coordinate system (or the **polar graph** of f) is the set of all points (r, θ) where r = f(θ)), or equivalently, (f(θ), θ) for θ in the domain of f.

When **graphing** a **function** in the **Cartesian** coordinate system, it is customary to use x and y and write y = f(x); and when **graphing** in the **polar** coordinate system, it is customary to use r and θ and write r = f(θ).

**Remark**: : It is customary to write the ordered pair (f(θ), θ) as (r(θ), θ) just as (x, f(x)) can be written as (x, y(x)).

The following figures created by the graphing calculator show the **graphs** of some **functions** drawn in the **Cartesian** and **polar** coordinate systems.

The graphs of the same functions as above drawn with axes rotated in each coordinate system.