# 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 se**t 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.

- With a graph:

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

**range**or

**image**is the subset of the codomain of the elements that are image of an element of the domain.

**N**and the range is the set of even numbers.

it is not a function

Domain = R

Range = [-3,3]

Domain = R Range = Z

**Exercise**: decide if the following relations are functions or not and, if they are, calculate its domain and range:

a)

b)

c)

Solutions: a) it is a function; domain = **R**; range = [1,∞); b) it isn't a function; c) it is a function; domain = **R**; range = [0,1)

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