# Understanding Functions in Calculus: A Comprehensive Guide

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In calculus, one of the most essential concepts is the function. The term 'function' was first introduced by **Gottfried Wilhelm Leibniz** in 1673 to describe the “functional” attributes of mathematical curves.

Over time, this idea has developed and become a core element not only in calculus but throughout mathematics as a whole. This article aims to delve deeply into the understanding of functions.

To begin with, I will outline the traditional application of functions in continuous mathematics, highlighting key technical characteristics and terminologies associated with them.

Next, I will extend the definition of a function into the realm of discrete or discontinuous mathematics. Finally, I will generalize how functions are utilized in contemporary mathematics.

I have observed that for various technical and rhetorical reasons, many young learners are not exposed to the complete essence of functions. As they progress into more advanced studies, they often rely on computational tools to manage 'functions' without grasping their entire scope (pun intended).

Thus, the objective of this article is to bridge this gap. Let us proceed without delay.

## The Fundamentals of a Function in Traditional Calculus

Historically, a function was simply defined as a relationship between two **variables**. When discussing 'variables,' there are a couple of conventions that might interest you.

In mathematics, it is customary to use letters from the end of the English alphabet (x, y, z, etc.) to signify variables, while letters from the beginning (a, b, c, etc.) represent **constants**.

For instance, consider the equation of a line:

y = ax + b

In this context, 'x' and 'y' are variables, whereas 'a' and 'b' are constants. This example also illustrates another convention closely tied to 'functions.'

Here, 'x' typically denotes the **independent variable**, while 'y' represents the **dependent variable** (whose value relies on that of x). For example, assuming the constants (a = 1) and (b = 2), the value of 'y' is determined by 'x' as follows:

y = f(1) = 1*(1) + 2 = 3

y = f(2) = 1*(2) + 2 = 4

y = f(3) = 1*(3) + 2 = 5

You may recognize the 'f(x)' notation in the above expressions. This stems from the convention that the dependent variable can be expressed as a 'function of' the independent variable.

The aforementioned expressions exemplify an explicit function of x. Here’s how you can express it implicitly using the line equation:

y - ax - b = 0

This can easily be transformed into the explicit form through algebraic manipulation.

## Further Exploration and Examples

Here’s another function example:

y = x²

Assuming 'y' represents the area of a square and 'x' its side length, this function is also a **one-to-one function** since the relationship between the variables is bidirectional.

However, maintaining the convention where 'x' is independent and 'y' is dependent, the reversed equation appears as follows:

y = ?x

where y = side length of the square and x = area of the square.

Note that functions do not necessarily exhibit a one-to-one relationship. For example, in a right triangle with sides 'x' and 'y' (which may not be equal), and hypotenuse 'z', the Pythagorean theorem allows us to calculate the hypotenuse using the following equation:

z = f(x, y) = ?(x² + y²)

Here, knowing the values of the sides 'x' and 'y' allows us to uniquely determine the hypotenuse 'z'. However, knowing 'z' does not allow us to uniquely ascertain 'x' and 'y'.

Moreover, the hypotenuse is also a function of two variables, referred to as a **two-variable function**.

A function can accommodate any number of independent variables. An example of a three-variable function would be the equation for the volume of a three-dimensional room defined by its length, width, and height.

Similarly, a four-variable function might describe the volume of a **four-dimensional hyper-room**.

## Visualizing Functions on the Cartesian Plane

As the concept of a function has evolved, mathematicians have come to agree that for a 'relation' to qualify as a valid function, each possible value of the independent variable *must uniquely correspond to a single dependent variable value only*.

To illustrate this crucial point, consider the function [y = f(x) = x²] plotted on the Cartesian plane. The two-dimensional Cartesian plane (named after its creator **René Descartes**) features a horizontal x-axis, a vertical y-axis, and an origin where the axes intersect.

Notice that each x-value is linked to a unique y-value along the function curve. If a vertical line drawn from any point on the x-axis intersects the plotted curve at more than one point, then the curve does not represent a function.

For instance, here is a curve that does not qualify as a function:

However, the same value of the dependent variable (y) may correspond to different x-values, *but not vice versa*. The subsequent curve exemplifies a valid function.

## Continuity, Domain, and Range

The Cartesian plane serves as a valuable tool for visualizing continuous functions. The branch of mathematics that deals with these continuous functions is aptly termed *continuous mathematics*.

A high-school teacher might define a two-dimensional continuous function as:

“A two-dimensional function is continuous if you can plot its curve on a two-dimensional Cartesian plane using a pen/pencil without lifting your hand.”

While this is not the strict mathematical definition of continuity, it suffices for the purposes of this article. A more rigorous definition will be covered in a future piece if necessary.

With the emergence of function analysis as a sub-field and the visualization on the Cartesian plane, mathematicians felt compelled to further generalize the definition of a function. This led to the introduction of the terms **Domain** and **Range**.

The 'Domain' encompasses all possible values that the independent variable (x-axis) can assume, while the 'Range' includes all possible values that the dependent variable (y-axis) can take. Both the domain and range may be bounded, unbounded, or intermittent.

That is to say, either can represent finite or infinite sets of values. In most traditional calculus contexts, these sets depict continuous intervals. There are indeed more complex functions, but delving into them is unnecessary for now.

Having discussed the traditional perspective on functions that most students are likely familiar with, I will now shift to a viewpoint that receives less attention.

## The Fundamentals of a Function Beyond Calculus

Recall that a one-variable function signifies a relation between an independent variable and a dependent variable, where each independent variable maps uniquely to a single dependent variable value.

Here’s an example of such a function from set theory:

What distinguishes this 'relation' is that *it is not defined by an equation* (as seen in calculus). Instead, the rules are arbitrarily established through mapping arrows (commonly known as **correspondence rules**).

This represents a significant departure from the relatively straightforward concepts previously discussed. Suddenly, the regulations governing a function can be arbitrary, provided its fundamental structure is maintained.

However, mathematicians are ever-curious and sought to expand their understanding!

## The Generalized Concept of a Function

With respect to functions, someone posed the following question:

“What if we generalize the notion of a function beyond mere numbers?”

This inquiry sparked a revolution in the understanding of functions, leading to disagreements along the way.

Ultimately, mathematicians broadened the definition of a function to the extent that *it is now often treated as a black box that adheres to the original structure of a function*.

In a conventional sense, a mother may have several children, but a child can have only one mother. Thus, mothers can be viewed as a function of children. However, grandmothers are not a function of grandchildren, as grandchildren can have multiple grandmothers.

Similarly, the geographic locations of towns can be considered a function of their positions on Earth.

The logical leap from numbers to mothers, grandmothers, and towns may seem subtle, but the mathematical implications are profound!

This generalized concept of the 'black box' function leads to numerous extremes that extend beyond the scope of this article.

For now, it is sufficient for readers to understand that the generalized notion of functions transcends continuous mathematics and numbers, *continuing to push the boundaries of mathematics even today*!

**References and Acknowledgments:** Martin Gardner, J.J. O’Connor, and E.F. Robertson.

I have published a follow-up article covering the foundational aspects of limits.

If you wish to support me as an author, **consider clapping, following, and subscribing.**

Further readings that may pique your interest include:

Is ‘0.99999…’ Really Equal To ‘1’?

How To Benefit From Computer Science In Real Life (I)

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*You can read the original essay here.*