Representations of functions I

Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations.

This post includes a draft of the introduction to the new chapter (immediately below) and of the section Graphs of continous functions of one variable. Later posts will concern multivariable continuous functions, probably in two or three sections, and finite discrete functions.

Introduction to the new abstractmath chapter on representations of functions

Functions can be represented visually in many different ways. There is a sharp difference between representing continuous functions and representing discrete functions.

For a continuous function $f$, $f(x)$ and $f(x’)$ tend to be close together when $x$ and $x’$ are close together. That means you can represent the values at an infinite number of points by exhibiting them for a bunch of close-together points. Your brain will automatically interpret the points nearby that are not represented.

Nothing like this works for discrete functions. As you will see in the section on discrete functions, many different arrangements of the inputs and outputs can be made. In fact, different arrangements may be useful for representing different properties of the function.

Illustrations

The illustrations were created using these Mathematica Notebooks:

These notebooks contain many more examples of the ways functions can be represented than are given in this article. The notebooks also contain some manipulable diagrams which may help you understand the diagrams. In addition, all the 3D diagrams can be rotated using the cursor to get different viewpoints. You can access these tools if you have Mathematica, which is available for free for faculty and students at many universities, or with Mathematica CDF Player, which runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

Graphs of continous functions of one variable

The most familiar representations of continuous functions are graphs of functions with one real variable. Students usually first see these in secondary school. Such representations are part of the subject called Analytic Geometry. This section gives examples of such functions.

There are other ways to represent continuous functions, in particular the cograph and the endograph. These will be the subject of a separate post.

The graph of a function $f:S\to T$ is the set of ordered pairs $\{(x,f(x))\,|\,x\in S\}$. (More about this definition here.)

In this section, I consider continuous functions for which $S$ and $T$ are both subsets of the real numbers. The mathematical graph of such a function are shown by plotting the ordered pairs $(x,f(x))$ as points in the two-dimensional $xy$-plane. Because the function is continuous, when $x$ and $x’$ are close to each other, $f(x)$ and $f(x’)$ tend to be close to each other. That means that the points that have been plotted cause your brain to merge together into a nice curve that allows you to visualize how $f$ behaves.

Example

This is a representation of the graph of the curve $g(x):=2-x^2$ for approximately the interval $(-2,2)$. The blue curve represents the graph.

The brown right-angled line in the upper left side, for example, shows how the value of independent variable $x$ at $(0.5)$ is plotted on the horizontal axis, and the value of $g(0.5)$, which is $1.75$, is plotted on the vertical axis. So the blue graph contains the point $(0.5,g(0.5))=(0.5,1.75)$. The animated gif upparmovie.gif shows a moving version of how the curve is plotted.

Fine points

• The mathematical definition of the graph is that it is the set $\{(x,2-x^2)\,|\,x\in\mathbb{R}\}$. The blue curve is not, of course, the mathematical graph, it represents the mathematical graph.
• The blue curve consists of a large but finite collection of pixels on your screen, which are close enough together to appear to form a continuous curve which approximates the mathematical graph of the function.
• Notice that I called the example the “representation of the graph” instead of just “graph”. That maintains the distinction between the mathematical ordered pairs $(x,g(x))$ and the pixels you see on the screen. But in fact mathe­maticians and students nearly always refer to the blue line of pixels as the graph. That is like pointing to a picture of your grandmother and saying “this is my grandmother”. There is nothing wrong with saying things that way. But it is worth understanding that two different ideas are being merged.

Discontinuous functions

A discontinuous function which is continuous except for a small finite number of breaks can also be represented with a graph.

Example

Below is the function $f:\mathbb{R}\to\mathbb{R}$ defined by
f(x):=\left\{ \begin{align} 2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\gt0) \\ 1-x^2\,\,\,\,\,\,(-1\lt x\lt 0) \\ 2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\lt-1) \end{align}\right.

Example

The Dirichlet function is defined by
$F(x):= \begin{cases} 1 & \text{if }x\text{ is rational}\\ \frac{1}{2} & \text{if }x\text{ is irrational}\\ \end{cases}$  for all real $x$.

The abmath article Examples of functions spells out in detail what happens when you try to draw this function.

Graphs can fool you

The graph of a continuous function cannot usually show the whole graph, unless it is defined only on a finite interval. This can lead you to jump to conclusions.

Example

For example, you can’t tell from the the graph of the function $y=2-x^2$ whether it has a local minimum (because the graph does not show all of the function), although you can tell by using calculus on the formula that it does not have one. The graph looks like it might have a vertical asymptotes, but it doesn’t, again as you can tell from the formula.

by looking at its graph
is useful but dangerous.

Example

Below is the graph of the function
$f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}$

If you didn’t know the formula for the function (but know it is continuous), you could still see that it has a local maximum somewhere to the right of $x=1$. It looks like it has one or more zeroes around $x=-1$ and $x=2$. And it looks like it has an asymptote somewhere to the right of $x=2.5$.

If you do know the formula, you can find out many things about the function that you can’t depend on the graph to see.

• You can see immediately that $f$ has a zero at $x=\sqrt[3]{10}$, which is about $2.15$.
• If you notice that the denominator is positive for all $x$, you can figure out that
• $\sqrt[3]{10}$ is the only root.
• $f(x)\geq0$ for all $x$.
• $f$ has an asymptote as $x\to-\infty$ (use L’Hôpital).
• Numerical analysis (I used Mathematica) shows that $f'(x)$ has two zeros, at $\sqrt[3]{10}$ and at about $x=1.1648$. $f”(1.1648)$ is about $-10.67$ , which strongly suggests that $f$ has a local max near $1.1648$, consistent with the graph.
• Since $f$ is defined for every real number, it can’t have a vertical asymptote anywhere. The graph looks like it becomes vertical somewhere to the right of $x=2.4$, but that is simply an illustration of the unbelievably fast growth of any exponential function.
• The section on Zooming and Chunking gives other details.

Acknowledgments

Sue VanHattum.