Domain, co-domain and image

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Domain, co-domain and image there are three different sets related to the study of a function. So, to understand what these sets are, we need to understand, first, what a function is.

Occupation is a set of ordered pairs (x, y), where each value of x is related to one, and only one, of the values ​​of y, through a formation rule: y = f(x).

Function example
Representation of a function.

Examples of functions and non-functions:

Examples of Functions and Non-Functions

Now that we know what is and isn't a role, let's look at the domain, counterdomain, and image definitions.

What is domain, counter-domain and image


It is the set formed by all values ​​of the variable x, for which the function exists, that is, those that have one, and only one, associated y-value.

Abbreviation: Sun (f).


It is the set formed by all the values ​​that variable y can assume, that is, that may or may not be associated with the values ​​of variable x.

Abbreviation: CD(f).


It is a subset formed by all values ​​of the counterdomain that have an association with some of the elements of variable x.

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Abbreviation: Im (f).

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Domain, co-domain and image
Domain, counter-domain and image representation.

Example: Consider the sets X = {0, 1, 2, 3} and Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and the function defined by the following rule:

f: X → Y

y = f (x) = 3x

We have:

Domain: D(f) = X = {0, 1, 2, 3}.

Counterdomain: CD(f) = Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Image: Im (f) = { f (0), f (1), f (2), f (3) } = {0, 3, 6, 9}, because:

f (0) = 3.0 = 0

f(1) = 3. 1 = 3

f(2) = 3.2 = 6

f(3) = 3.3 = 9

To be a function, all domain elements must have one, and only one, corresponding element in the counterdomain. Note that this happens in the function above.

However, it is not necessary that all elements of the counterdomain have a counterpart in the domain. See, for example, that the values ​​1, 2, 4, 5, 7, 8, and 10 of set Y have no association with any value of X.

You may also be interested:

  • First degree function (affiliated function)
  • First degree function exercises (affine function)
  • Trigonometric Functions - Sine, Cosine and Tangent

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