Produced by Charles Wells Revised 2017-03-08 Introduction to this website website TOC website index blog head of Functions chapter

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

- This article concentrates on
**visual representations of continuous functions**. Functions can also be represented by**formulas**and**algorithms**; some examples are given in Functions: Images and Metaphors. - 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 a*finite*bunch of points that are so close together they appear to merge into a curve. This gives the "graph" in the sense used in precalculus and calculus classes. - A graph of a continuous real-valued function such as the one below is necessarily a non-precise representation of the function. For example, you cannot measure the graph at a point to determine the exact value at that point.

As you will see in Representations of Discrete Functions, many different arrangements of the inputs and outputs for discrete functions can be made, but none of them can use the idea that close inputs give close outputs. On the other hand, a representation of a finite function can be absolutely accurate.

The illustrations were created using these Mathematica Notebooks:

- ContinuousFunctionsForFuncRepsTwo.nb
- Functions and Shapes.nb
- These notebooks contain many more examples of the ways functions can be represented than are given in this article.
- The section Mathematica in the abmath Introduction explains how to use the notebooks and CDF files.
- Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

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.

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

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 plotparmovie.gif shows a moving version of how the curve is plotted.
- The CDF file uppar.cdf also shows the way the curve is plotted in a way that you can manipulate.

- 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. - I called the blue curve 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. In fact mathematicians and students nearly always refer to the 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.

A function which is continuous except for a small finite number of breaks can also be represented with a graph. Such functions are called piecewise continuous functions.

Below is a graph of (part of) the function $f$ defined by\[f(x):=\left\{
\begin{align}
2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\gt0) \\
1-x^2\,\,\,\,\,\,(-1\lt x\lt 0) \\
2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\lt-1)
\end{align}\right.\] $f$ is *one function*. It is defined for every real number except $-1$ and $0$.

Another discontinuous function is shown in Examples of functions

This example uses the concepts of rational and irrational.

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 $x$.

It is impossible to draw a graph of this function, because it has an infinite number of gaps between its points. For more detail, see the article Examples of functions.

The graph of a continuous function cannot usually show the whole graph, unless it is defined only on a finite interval. It is also only approximately accurate: You cannot determine precisely the exact location of any point on the graph. These faults can lead you to jump to incorrect conclusions.

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 also looks like it might have vertical asymptotes, but it doesn't, again as you can tell from the formula (it is defined for all real numbers).

Discovering facts about a function

by looking at its graph

is useful but dangerous.

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**, by looking at the graph you might nevertheless suspect certain things are true of the function.

- Maybe it has a local maximum somewhere to the right of $x=1$.
- Maybe it has one or more zeroes around $x=-1$.
- Maybe it has one or more zeroes around $x=2$.
- Maybe it has a vertical asymptote somewhere to the right of $x=2.5$.
- Maybe it has a horizontal asymptote in the negative $x$ direction.

**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 just one zero, at $x=\sqrt[3]{10}$, which is about $2.15$. So (3) is correct.
- If you notice that the exponent $6$ is even, you can figure out that $f(x)\geq0$ for all $x$, so $\sqrt[3]{10}$ is an absolute minimum for $f$.
- By using L'Hôpital several times, you can see that $f$ has a horizontal asymptote as $x\to-\infty$. So (5) is correct.
- 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 shows that $f$ has a local max near $1.1648$, consistent with the graph. So (1) is correct.
- The fact that $f'(x)$ has only two zeros, both for positive values of $x$, means that there can be no minimum for negative $x$. So (2) is incorrect.
- Since $f$ is defined for every real number,
*it can't have a vertical asymptote anywhere*. So (4) is incorrect. 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.

Suppose $F:\mathbb{R}\to\mathbb{R}\times\mathbb{R}$. That means you put in one number and get out a pair of numbers.

An example is the **unit circle**, which is the graph of the continuous function $t\mapsto(\cos t,\sin t)$. That has this **parametric plot**:

Because $\cos^2 t+\sin^2 t=1$, every real number $t$ produces a point on the unit circle. Four point are shown. For example,\[(\cos\pi,\,\sin\pi)=(-1,0)\] and \[(\cos(5\pi/3),\,\sin(5\pi/3))=(\frac{1}{2},\frac{-\sqrt3}{2})\approx(.5,-.866)\]

In graphing functions $f:\mathbb{R}\to\mathbb{R}$, the plot is in two dimensions and consists of the points $(x,f(x))$: the input and the output. The parametric plot shown above for $t\mapsto(\cos^2 t+\sin^2 t)$ shows only the output points $(\cos t,\sin t)$; $t$ is not plotted on the graph at all. So the graph is in the plane instead of in three-dimensional space.

It is useful to think of $t$ as **time**. If you start at some number $t$ on the real line and continually increase it, the value $f(t)$ moves around the circle counterclockwise, repeating every $2\pi$ times. If you decrease $t$, the value moves clockwise. Every point is traversed an infinite number of times as $t$ runs through all the real numbers.

- The animated gif circlemovie.gif shows how the location of a point on the circle moves around the circle as $t$ changes from $0$ to $2\pi$.
- The CDF file circlepar.cdf shows the movement of the point as in the movie, but with more control over the behavior of the point.

This is the parametric graph of the function $t\mapsto(\cos t,\sin 2t)$.

Notice that it crosses itself at the origin when $t$ is any odd multiple of $\frac{\pi}{2}$.

The graph of a function from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ can also be drawn as a parametric graph in *three*-dimensional space, giving a three-dimensional curve.

Here is a view of the graph of the function \[t\mapsto(\cos t, \sin t, \sin 7t):\mathbb{R}\to \mathbb{R}\times\mathbb{R}\times\mathbb{R}\]

The animated gif crownmovie.gif represents the parameter $t$ in time.

Since we have access to three dimensions, we can display on the graph the input $t$ to the unit circle function $(\cos t,\sin t)$ by using a **three-dimensional graph**, shown below. The blue circle is the function $t\mapsto(\cos t,\sin t,0)$ and the gold helix is the function $t\mapsto(\cos t,\sin t,.2t)$.

The introduction of $t$ as the value in the vertical direction changes the circle into a helix. The animated .gif covermovie.gif shows both the travel of a point on the circle and the corresponding point on the helix.

As $t$ changes, the circle is drawn over and over with a period of $2\pi$. Every point on the *circle* is traversed an infinite number of times as $t$ runs through all the real numbers. But each point on the *helix* is traversed exactly once. For a given value of $t$, the point on the helix is always directly above or below the point on the circle.

The helix is called the universal covering space of the circle, and the set of points on the helix over (and under) a particular point $p$ on the circle is called the **fiber over $p$**. The universal cover of a space is a big deal in topology.

Here are two views of the function \[t\mapsto(\cos t,\sin 2t, \log t):\mathbb{R^{+}}\to \mathbb{R}\times\mathbb{R}\times\mathbb{R}\] ($\mathbb{R}^{+}$ is the set of positive real numbers)

It is a universal cover (with an added squish as $z$ increases) of the Figure-8 curve.

- For the second picture, talk yourself into seeing the top as closer to you than the bottom. I am not kidding.
- The $x$ and $y$ coordinates are those of the Figure 8 construction. And indeed the curve swirls around in a rising figure-8 like a figure skater on a rising platform.
- The swirls get closer together as you go up the $z$-axis. That is because $\log t$ increases more and more slowly as $t$ gets bigger:

\[z=\log t\]

The graph of a map $\mathbb{R}^m\to\mathbb{R}^n$ where $m+n\gt3$ requires $m+n$ dimensions, so we can't draw it and it is difficult to visualize it.

You *can* graph a function $\mathbb{R}\times\mathbb{R}\to\mathbb{R}\times\mathbb{R}$ using its endograph.

The function \[(x,y)\mapsto(x+.5,y+.2)\] moves each point in the plane $.5$ units to the right and $.2$ units up. The endograph below visualizes a part of this action. showing the movement at each point with integer coordinates.

This is an example of an affine map in the plane.

The linear map determined by the matrix \[\pmatrix{1.25 & 0.5\\0.5&1}\] defines a function \[(x,y)\mapsto (1.25x+.5y,\,.5x+y):\mathbb{R}\times \mathbb{R}\to \mathbb{R}\times \mathbb{R}\] that can be displayed with an endograph like this:

- This graph shows arrows pointing to the value for integer pairs within the range. Because the function is continuous, you can guess approximately where a point goes by where the nearby integer points go. That is not possible for discrete functions.
- Vector fields are often illustrated in this way. In a vector field, an arrow shows the direction and velocity of movement at a given point, whereas, in the graph above, an arrow
*points to the value of the function*at an integer pair $(m,n)$. - MÃ¶bius transformations are an important type of function from $\mathbb{R}\times \mathbb{R}$ to itself. This video on YouTube illustrates them visually using another technique.

Complex functions are often graphed using $\mathbb{R}\times\mathbb{R}$ for the domain (representing $a+bi$ as $(a,b)$), with the values of the function shown as hue and brightness, as seen in the Wikipedia article.

Complex analysis, the theory of functions on the complex numbers, is one of the richest and most highly developed parts of math.

This is the graph of part of the surface\[(x,y)\mapsto(x,z,\sin xz):\mathbb{R}\times\mathbb{R}\to\mathbb{R}\times\mathbb{R}\times\mathbb{R}\]

The blue curves follow the shape of the surface as an aid to visualizing the surface.

This graph was created by the Mathematica notebook AnotherFamiliesFrozen.nb It is worthwhile viewing this notebook in CDF Player, if you have it. (It is free). There are more examples in Freezing a family of functions.

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