Two Faces of Functions
Functions appear in many ways and we can try to understand them
in many ways. Here, we will discuss two important ways of discussing
functions which you will see in this course.
functions through measurements
The graph below represents an interesting function from a famous
event this summer. When the Mars Pathfinder arrived at Mars, it
needed to slow down or brake so that it wasn't destroyed by the impact
with the planet. The graph shows the relationship between the
deceleration (or rate of braking) and time. The large bump in the
graph occurred when Pathfinder entered the Martian atmosphere. The
second, smaller spike occurred when the parachutes were deployed.
The appearance of this graph may make you think that we actually
know the deceleration at every time. But don't be fooled by this
appearance. The graph was built from regular reports of the
deceleration dispatched from Pathfinder; that is, Pathfinder told us
the deceleration every so often (maybe every second). This
information gave data points on the graph through which a curve was
drawn to give a smooth appearance. In fact, if we were to look very
closely at a piece of the graph, it probably looks something like:
Still we feel fairly confident that
the picture, which is extrapolated from the data points, reflects the
actual function quite accurately.
The main point here is that we only really know what happens at a
certain number of points but, based on that knowledge, we may infer
what happens at all times. For scientists, this is one of the most
practical ways for thinking about functions since it is how they are
often encountered---through measurement.
For your consideration:
If you were one of the designers of Pathfinder,
why would you want to understand this function before the mission
began? What kind of information would you want to know about it?
Now we'll talk about a different way of describing some
functions which you have probably seen in earlier Mathematics courses.
For instance, we may refer to a particular function
By this, we mean that some quantity, which we call
depends on another quantity, which we call
We don't yet have any
but let's not worry about that just now.
We are, however, told quite specifically how y depends on
given value of
value is just the square of that
value. For instance,
of this function is shown below.
The main thing to notice is that this function tells us what
is for every value of
In the discussion above, we
only knew what happened at the points where we made measurements and
then guessed at what happened in between. So in some sense, the
notion of the function
is idealized since it assumes
that we have measured
at every value of
(and this would take a
very long time indeed!). However, we will convince you that this
kind of function is also very useful for scientists.
For your consideration:
Do you think there is a simple algebraic
Pathfinder's deceleration? Would it be useful to find one or is it
good enough to think about the function through its graph above?