What is a Function?

Theory

In mathematics, a function is a fundamental concept that describes a relationship between two sets of objects, known as the domain and the codomain. A function assigns an output value to each input value from the domain.

Formally, a function can be defined as follows:

Given two sets, X and Y, a function f from X to Y is a rule that assigns each element in X (domain), called the input, to an element in Y (codomain), called the output. It is denoted as:

f: X → Y

It is important to note that functional dependencies are always unambiguous: only one value from codomain Y can be assigned to a value from domain X. But simultaneously, several values from X can correspond to one value from Y.

Example 1

Let's look at the example to understand it. Assume that you have two points at the plane and want to calculate the Euclidean distance between these points. To do it, we use the distance function:

We have a domain consisting of coordinates of points and a codomain representing distances. The Euclidean function assigns a distance to each pair of points in the domain. It can be only one distance between two particular points, but a set of points can have equal distances between each other.

Example 2

When an object is launched into the air and follows a curved path under the influence of gravity, the trajectory of the object can be modeled by a parabolic function. The height (y) of the object at a given horizontal distance (x) can be represented by the equation of a parabola:


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import numpy as np
import matplotlib.pyplot as plt

# Define the parabolic function
def parabola(x):
    return -0.5 * x**2 + 3 * x + 5

# Generate x values
x = np.linspace(0, 6, 100)

# Calculate y values using the parabolic function
y = parabola(x)

# Plot the parabola
plt.plot(x, y)
plt.xlabel('Horizontal Distance (x)')
plt.ylabel('Height (y)')
plt.title('Projectile Motion: Parabolic Trajectory')
plt.grid(True)
plt.show()

For each x, there is only one assigned y value. But the converse is false: the same y can be assigned to different x values.

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Section 1. Chapter 1