Functions in mathematics

A function $ f $ is a relationship between two sets that assigns each element of the first set (domain) to exactly one element of the second set (codomain).

Formally, a function is denoted as $ f: A \rightarrow B $, where $ f $ is the function, $ A $ is the domain, and $ B $ is the codomain.

Domain: The set of all possible input values that the function can accept.
Codomain: The set of all possible output values that the function can produce.

In notation, a function can be written as $ f(x) $, where $ x $ represents an element of the domain and $ f(x) $ (or $ y $) is the element of the codomain associated with it.

$$ y = f(x) $$

The variable $ x $ is the independent variable, whereas $ y $ is the dependent variable as its value depends on the value of $ x $.

Once a value is assigned to $ x $, the value of $ y $ is uniquely determined by the formula that links $ x $ to $ y $. For instance, if the function $ f(x) $ is given by $ y = 2x $, then for $ x = 1 $, the value of $ y $ is $ y = 2 $. Similarly, for $ x = 2 $, we get $ y = 4 $, and so on.

Example
Types of Functions

Example

Consider the function $ y = f(x) $, where $ x $ is any real number.

$$ f(x) = 3x + 1 $$

This function assigns each number $ x $ a corresponding number $ y = f(x) $, calculated by multiplying $ x $ by 3 and then adding 1.

For instance, if $ x = 2 $, then:

$$ f(2) = 3(2) + 1 = 7 $$

In this case, the value $ y = 7 $ is the image of $ x = 2 $ under the function $ y = 3x + 1 $.

Here is a table showing values of the function $ f(x) = 3x + 1 $ for some selected values of $ x $ from the domain.

$$
\begin{array}{c|c}
x & y \\
\hline
-2 & -5 \\
-1 & -2 \\
0 & 1 \\
1 & 4 \\
2 & 7 \\
\end{array} $$

In this table, $ y $ represents the value of the function $ f(x) = 3x + 1 $ for each value of $ x $.

By plotting these data points on a coordinate plane, we can create the graph of the function $ f(x) = 3x + 1 $.

Let's plot these points on a Cartesian plane: $(-2, -5)$, $(-1, -2)$, $(0, 1)$, $(1, 4)$, $(2, 7)$

example of graph construction

Once these points are plotted on the Cartesian plane, we can connect them with a straight line since the function $ f(x) = 3x + 1 $ is linear.

the graph of the function

Types of Functions

Functions can be classified in various ways based on their properties. Here is an overview of the main categories:

Injective Functions

A function $ f: A \rightarrow B $ is injective if distinct elements of $ A $ are mapped to distinct elements of $ B $.

This means that each input value $ x $ produces a unique and unambiguous output value $ y $.

example of an injective function

So, if the function returns the same result $ f(x_1) = f(x_2) $ for two values $ x_1 $ and $ x_2 $ from the domain, this implies that $ x_1 = x_2 $.

Surjective Functions

A function $ f: A \rightarrow B $ is surjective if every element of $ B $ is the image of at least one element of $ A $.

In other words, the codomain is completely covered by the function, meaning each element $ y $ in the codomain $ B $ is associated with at least one element $ x $ in the domain $ A $.

the surjective function

Bijective Functions

A function is bijective if it is both injective and surjective. This implies a one-to-one correspondence between the elements of the domain and those of the codomain.

Every element of $ A $ is mapped to a unique element of $ B $ and vice versa.

example of a bijective function

Injective, surjective, and bijective functions represent the main categories that help classify and better understand the behavior of functions in mathematical applications.