Relating these representations of functions
UBC Calculus Online Course Notes

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.

Understanding 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?

Algebraically representing functions

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 $  y(x) = x^2. $ By this, we mean that some quantity, which we call $  y,  $ depends on another quantity, which we call $  x.  $ We don't yet have any interpretation for $  x  $ and $  y,  $ but let's not worry about that just now. We are, however, told quite specifically how y depends on $  x:  $ for a given value of $  x,  $ the corresponding $  y   $ value is just the square of that $  x   $ value. For instance, \[ 
y(1) = 1, ~~~~y(2) = 4, ~~~~y(-2) = 4. \] The graph of this function is shown below.

The main thing to notice is that this function tells us what $  y  $ is for every value of $  x.  $ 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 $  y = x^2  $ is idealized since it assumes that we have measured $  y  $ at every value of $  x  $ (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 expression---something like $  y = x^2  $ ---which describes Pathfinder's deceleration? Would it be useful to find one or is it good enough to think about the function through its graph above?