## 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.