Connecting the Dots

Date: 02/02/98 at 13:52:58
From: Kristine Freeman
Subject: Functions

Hello -

This particular question came up in my math class and I was wondering 
if you could explain the answer to me. My instructor was unsure of the 
answer also.

Some graphs of functions consist of dots. Others are lines or curves.  
How do you know whether or not to connect the dots when graphing a 
real-life function?

Thank you,
Kristine Freeman

Date: 02/10/98 at 16:14:27
From: Doctor Nick
Subject: Re: Functions

Hello Kristine -

This is a very good question.  

I think that what your question gets at is the concept of continuity.  
A function is continuous if its graph can be drawn without taking your 
pencil off the paper. Well, that's the intuitive idea. A graph that's 
made up of isolated dots, or separate pieces is called discontinuous.  

Now, there are ways to tell if a function is continuous. The study of 
continuity is a large part of the mathematical area called calculus.  
With calculus, you can show that functions that are polynomials are 
continuous. Also, functions that are rational, that is, that are 
quotients of polynomial functions, are also continuous, at least in 
the areas where they are defined.

The trigonometric functions sine and cosine are continuous. The 
exponential function e^x is continuous.

Many other functions are continuous. Many are not. Determining whether 
or not a function is continuous can be quite tricky, but calculus is 
the place to start. Even if you're not quite ready for a calculus 
class (I don't know what class you're in now), you might like to take 
a look at a calculus text. The topic of continuity comes up right near 
the beginning, very much connected to the starting concept of 
calculus, the limit.

An important factor to consider when graphing a function is the domain 
of that function. The domain is the set of values that can be plugged 
into the function so that it returns a real value. For instance, the 
domain of the function

   f(x) = 3*x

is the set of all real numbers, since 3 times any real number is a 
real number. Considering the graph of this function, we know the 
points (0,0) and (1,3) are on it. Since the function is a linear
function (we know that y=3*x is a straight line), we can "connect the 
dots" (0,0) and (1,3) since every x value between 0 and 1 is in the 
domain of the function, and so for every such x value there is a 
corresponding y value on the graph of f.  

Now, in the "real world" we sometimes run across functions for which 
the domain is not all real numbers, even though algebraically it 
should be. For instance, we might be told that the number of cars 
produced by a factory on a given day is c(x) = 3*x where x is the 
number of days from today. That is, today no cars are produced, 
tomorrow 3 cars and produced, 6 the day after that, then 9, etc.

Generally speaking, the function 3*x has the domain of all real 
numbers, but here we are given the extra condition that x is a whole 
number. The graph of this function then consists of the points
(0,0),(1,3),(2,6),(3,9),(4,12), etc. - isolated points. In this 
instance, it would not be correct to "connect the dots" since values 
of x that are not positive integers have no corresponding y value. 

Often "real-world" functions are presented as data rather than as 
algebraically expressed functions. We might know that on day 1, 3 cars 
were produced; on day 2, 7 cars; on day 3, 11 cars; on day 4, 10 cars; 
etc. These data define a function f(x) where x is a positive integer: 
f(1)=3, f(2)=7, f(3)=11, f(4)=10, etc. In this case again the graph is 
a bunch of isolated points, and connecting the dots would not give a 
correct graph.

However, for aesthetic (or other) reasons, people often do connect the 
dots of such graphs. This is implicitly creating a new function, say 
g(x), where g(x)=f(x) if x is a positive integer, and if x is not a 
positive integer, then g(x) is the y-value of the point (x,y) lying on 
the line connecting the two points on the graph (x1,f(x1)) and 
(x2,f(x2)) where x1 and x2 are the integers immediately less than
and greater than, respectively, x.

In all cases, remember that a graph (in the sense here) is a pictorial 
representation of a function. It is a way of communicating information 
about that function. Regardless of the method you use to create the 
graph, if the graph communicates the right information, then it's a 
good graph.

Have fun!

-Doctor Nick,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/