Calculus: How To Really Understand A Function

In calculus, arguably one of the most basic concepts is that of the function. The term ‘function’ was originally used for the first time by Gottfried Wilhelm Leibniz in 1673 to describe the “functional” properties of mathematical curves.

Over the years, this concept has evolved and expanded to become one of the most fundamental concepts in not just calculus but the whole of mathematics. In this essay, I will be diving into a deep discussion about how to really understand the notion of a function.

To this end, I will start by introducing its conventional use in continuous mathematics. Here, I will also be lingering upon some of the key technical characteristics and terms in relation to functions.

Then, I will proceed to expand the notion of a function to the world of discrete/discontinuous mathematics. Finally, I will try to generalise the concept as to how mathematics uses it today.

I’ve noticed that due to various technical and rhetorical reasons, young students are not exposed to the complete nature of functions. On the other hand, as students transform into advanced practitioners, they often get used to computational devices to handle ‘functions’ without having covered its entire domain (pun totally intended).

The aim of this essay is, then, to try and fill this gap. Without any further ado, let us begin.

This essay is supported by Generatebg

The Basics of a Function from Traditional Calculus

As far as Leibniz and co. were concerned, a function started as nothing but a relation between two variables. While we are on the topic of ‘variables’, there are a couple of points that might be of interest to you.

It is convention in mathematics to use letters from the end of the English alphabet (x, y, z, etc.) to refer to variables, whereas letters from the beginning of the English alphabet (a, b, c, etc.) are used to refer to constants.

For example, consider the equation equation of a line:

y = ax + b

Here, ‘x’ and ‘y’ are variables, whereas ‘a’ and ‘b’ are constants. This example also holds another subtle convention that is dear to ‘functions’.

You see, ‘x’ is often used to denote the independent variable, whereas ‘y’ is used to denote the dependent variable (whose value depends on x’s value). For instance, assuming the constants (a = 1) and (b = 2), the value of ‘y’ depends on the value of ‘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 might recognise the ‘f(x)’ from in the above expressions. This also comes from the convention that the dependent variable can be expressed as as a ‘function of’ the independent variable.

The above expressions are examples of an explicit function of x. Here is an example of an implicit function of x using the line equation:

y — ax — b = 0

You could easily arrive at the explicit form using algebraic manipulation.

More Basics and More Examples

Here is another example of a function:

y = x²

Here, let us assume that ‘y’ refers to the area of a square and ‘x’ refers to its side length. This function is also an example of a one-to-one function as the dependency of the variables goes both ways.

However, when we keep the convention that ‘x’ refers to the independent variable and ‘y’ refers to the dependent variable, the reversed equation looks as follows:

y = √x

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

Note that functions need not necessarily have a one-to-one relationship. Consider a right-angled triangle with sides ‘x’ and ‘y’ (not necessarily equal), and hypotenuse ‘z’. Using the Pythagorean theorem, the hypotenuse could be calculated using the following equation:

z = f(x, y) = √(x² + y²)

Here, knowing the value of the sides ‘x’ and ‘y’, we can uniquely determine the hypotenuse ‘z’. However, knowing the value of ‘z’, we cannot uniquely determine ‘x’ and ‘y’.

Furthermore, note that the hypotenuse is also a function of two variables. Such a function is known as a two-variable function.

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

Similarly, an example of a four-variable function would be the equation of the volume of a four-dimensional hyper-room. And so on.

---

Visualising Functions on the Cartesian Plane

As the notion of a function evolved through history, mathematicians agreed at some point (relatively recently) that for any ‘relation’ to be considered a valid function, each possible value of the independent variable should uniquely lead to/map with one dependent variable value only.

To practically illustrate this important point, consider the function [y = f(x) = x²] plotted on the Cartesian plane. The two-dimensional Cartesian plane (named after its inventor René Descartes) consists of a horizontal x-axis, a vertical y-axis, and an origin (representing zero) where the axes meet.

Note that each x-value is associated with a unique y-value as far as the function curve is concerned. Say that you draw a vertical line from any possible point on the x-axis. If that line cuts the plotted curve in more than one point, then the curve does not represent a function.

Here is an example of a curve that is not a function:

Notice, however, that the same value of the dependent variable (y) may be mapped to the different x-values. But not the other way around. The following curve, for instance, represents a valid function.

Continuity, Domain, and Range

The Cartesian plane is useful to visualise continuous functions. The mathematics that deals with continuous functions is appropriately called continuous mathematics.

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

“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.”

This, of course, is not the strict mathematical definition for continuity. For the purposes of this essay, however, it suffices. I’ll cover the strict definition in a later essay if the need arises.

With the rise of function-analysis as a sub-field and visualisation on the Cartesian plane, mathematicians felt the need to generalise the definition of a function some more. Thus, the terms ‘Domain’ and ‘Range’ were born.

‘Domain’ refers to all possible values that the independent variable (x-axis) can take, whereas ‘Range’ refers to all possible values that the dependent variable (y-axis) can take. Either the domain or the range or both can be bound/unbound/intermittent.

That is, either of them can take infinite/finite sets as possible values. For most of the traditional calculus, these sets represent continuous intervals. There are more complex functions, for sure. But there is no need to dive into them for now.

So far, I have been covering the traditional view of functions that most students must be familiar with. Now, I depart from this perspective to a perspective of functions that does not get as much attention.

---

The Basics of a Function from Outside Calculus

Recall that a one-variable function represents a relation between an independent variable and a dependent variable such that each independent variable is uniquely mapped to a single dependent variable value only.

Here is an example of such a function from set theory:

What is different about this ‘relation’ is that it is not defined by an equation (like we saw from the world of calculus). Instead, the rules are arbitrarily determined using the mapping arrows (commonly referred to as correspondence rules).

This is a stark departure from the relatively tame world we just came from. All of a sudden, the rules of a function can be arbitrary, as long as its fundamental structure is respected.

But the mathematicians are a greedy bunch. They did not stop there and wanted more!

The Generalised Notion of a Function

As far as the notion of a function is concerned, someone, somewhere, asked the following question:

“What if we generalise the notion of a function beyond just numbers?”

This led to a revolution in terms of what a function can be and cannot be. Naturally, such a revolution was not without disagreement.

Ultimately, though, the result was this: mathematicians generalised the definition of a function to the point that it is now treated pretty much as a black box that respects a function’s original structure.

In a conventional sense, a mother may have several children, but a child can have only one mother. In this way, mothers are a function of children. But grandmothers are not a function of grandchildren, because grandchildren can have more than one grandmother.

Similarly, the map locations of towns are a function of their geographic location on the Earth.

The logical jump from numbers to mothers, grandmothers, and towns might be subtle, but the mathematical jump is huge!

This notion of the generalised ‘black box’ function leads to numerous extremes that go beyond the scope of this essay.

For now, it suffices if the reader knows that the generalised notion of the function extends beyond just continuous mathematics and numbers and pushes the fringes of mathematics even today!

---

Reference and credit: Martin Gardner, J.J. O’Connor and E.F. Robertson.

If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you:

• Is ‘0.99999…’ Really Equal To ‘1’?
• How To Benefit From Computer Science In Real Life (I)
• What Is So Special About 69! ?

If you would like to support me as an author, consider contributing on Patreon.