# Functions

A** function** f, is a relation between two sets with the property that each element in the first set is related to exactly one element in the second set.

f: X →Y, X is the **initial set** and Y the **final set** (**codomain**)

x →y = f(x) x is called **independent variable** (variable) and y is called **dependent variable**. f(x) is **the image of x under f**.

Example 1: the function that associates each student of this class with his or her age.

Example 2: the function that associates each natural number with its double.

There are 4 ways to express a function:

- *Verbal way*, as in the examples above.

- *Algebraic way,* with a formula. In the example 2: y = f(x) = 2x

- *With a table*:

- *With a graph:*

The **domain** of a function is the subset of the initial set of the elements that have an image.

The **range or image** is the subset of the codomain of the elements that are images of an element of the domain.

Example 1: the initial set and the domain is the class, the codomain is the natural numbers and the range = {15, 16, 17}.

Example 2: the initial set and the domain is N and the range is the set of even numbers.

Example 3:

it is not a function

Example 4:

Domain = R

Range = [-3,3]

Example 5: Whole number portion function

Domain = R Range = Z

**Exercises**:

1.- Decide if the following relations are functions or not, and if they are, find their domain and range:

a)

b)

c)

Solutions:

a) it's a function, dom = [-1,2]U[3,5)U(5,8], range = [0,5]

b) it's a function, dom = range = R

c) it is not a function

## 2.- True-False Question

Decide if the following relations are functions:

#### Retroalimentación

**Falso**

#### Retroalimentación

**Falso**

#### Retroalimentación

**Verdadero**

Obra publicada con Licencia Creative Commons Reconocimiento No comercial Compartir igual 3.0