The Meanings of Functions
I know you are familiar with the common set of analytic functions used in mathematics. The list really isn't very long. With a few omissions, it mainly includes polynomials, trigonometric functions, and exponentials. I want to take a little time to discuss each in turn.
Polynomials
Equations of the form , where n is usually an integer value, are polynomials. The word polynomial comes from Greek and means many parts. This is useful because it also tells us what a polynomial function really means. It means that y depends on many factors or "parts".
As an example, the area of a square fits where L is the side length. In this sense the area depends on two different dimensions or "parts" of the square. Of course the volume depends on three "parts" and is obviously cubic in L. In this context it is useful to note that these "parts" should be independent in the sense that the width can grow without affecting the length and vice versa.
As another example, the force objects encounter as they move through our atmosphere at moderate speeds, which we call air drag, depends on speed squared. When an experiment is conducted and we see this quadratic dependence on speed, it actually tells us something. It tells us that there must be two independent aspects of the speed that cause air drag to increase. In the case of air drag the first is that each collision between the object experiencing drag and each air molecule becomes twice as violent if the speed is doubled (three times if speed is tripled, etc). If this were the only factor, however, then air drag would depend on speed to the first power. The other factor of speed in the drag force comes from the fact that air is a gas, and gasses have lots of space between molecules. In this sense the collisions experienced are not continuous, but more like being pelted by paint balls or something... except really small ones and lots of them.
The faster an object travels through air, the more collisions it experiences with air molecules, because it travels between the voids among the molecules in less time. Taken together, traveling at twice the speed would mean twice as violent collisions (one factor of speed) occurring twice as frequently (the other consequence of speed). This makes air drag quadratic. At double the speed it become four times larger. At triple the speed the collisions are three times as violent and occur three times as frequently and drag is therefore nine times greater.
As a last example, it is seen that certain forms of cancers arise in human populations with age to the 4th power. If you've studied a little biology you probably know that the body has multiple mechanisms to prevent cancer. The rise in cancer rates to the 4th power with age suggests that 4 independent aspects of biology must go wrong in time in order for such cancers to arise. It's a good thing that a simple DNA mutation is not sufficient since we get those from sunlight and our environment all the time!
It is useful to think about mathematical functions for what they suggest to us about the science rather than just seeing them as equations.
Trigonometric Functions
Your first introduction to trigonometry arose from studies of triangles. So naturally these functions have a lot to do with geometry and angles. In simple applications we should expect almost always that where angles are involved we will find trigonometric functions.
For example, given a vertical post of a known height in the ground, and the angle of incident sunlight, it is easy to find the length of the subsequent shadow.
Not long after studying triangles it was probably shown to you that triangles can be inscribed within a unit circle. This suggests that trigonometric functions have a lot to do with circles as well, at least when we use Cartesian coordinates. In that case we tend to find them in rotating systems all over nature. Furthermore we will discuss in a chapter on harmonic motion how rotation seen from a different perspective looks exactly like oscillation - or vibration.
This suggests that any time we have angles and geometry, things rotating or things oscillating we are likely to find sines, cosines and tangent functions and other related ones. We also tend to find them in any traveling wave-like disturbance. So we find them all over nature describing light waves, sound waves, and other wave-like phenomena as well.
Exponential Functions
First I want to say that I hope you never just look at a function that is growing faster than linearly and just decide to call it an exponential. After all, a parabola does this, and yet it is completely incorrect to call it exponential growth.
Exponential growth or decay is a very special situation. To be a true exponential function, a function must grow (rate of change) in direct proportion to the value of the function.
This is clear from calculus where you see that the derivative of an exponential is an exponential. In other words its rate of change is proportional to itself. No other function has this property.
When we see exponential behaviors in nature this gives important hints regarding mechanisms at work. For instance, we see that when food supply is abundant and freely available to all members of a culture, that bacterial populations grow exponentially. Suppose that, for instance, starting with a single bacterium, that it reproduces such that we have 2 bacteria after one minute. Now that we have two, each will double in the next minute giving us 4 after 2 minutes. Then we'd have 8 after 3 minutes, 16 after 4 minutes, etc. The rate of growth of the population () is proportional to the size of the population (N). This is exponential growth. It cannot proceed like this forever. They will exhaust the food supply eventually, and the system will be limited. Nonetheless, in early stages of infection, populations of bacteria grow in this exponential way. So do populations of animals in the absence of predators, death, and with a large food supply.
Another example is nuclear decay. When a nucleus decays it does so independently of the other nuclei. We know this because nuclear decay occurs in an exponential fashion. The decay rate only depends on the size of the population. Just as bacteria multiply in proportion to the size of the population, so nuclei decay in proportion to the size of the sample.
To show this relationship, here is an example of radioactive decay. Suppose the number of radioisotopes is N0 to begin with, and we want to know how many exist (N) at time t. The decay rate is proportional to the size of the population. The decay rate is the proportionality constant, and the population shrinks in time since the nuclei are decaying. Mathematically this looks like:
As a last example, there is a type of friction in nature called viscous damping. In such cases, as when objects move through viscous fluids like water at slower speed, or through really viscous fluids like honey, the rate of change of speed loss from the friction is proportional to the speed itself. This will naturally lead to speed's plot over time being an exponential decay for the very same reasons discussed above about the nature of exponential functions. This turns out to be the case because viscous drag depends only linearly on the speed. In this case, doubling the speed doubles the viscous drag force, which as we'll see soon leads to a doubling of the rate of speed loss.
Why We Care
We should care to know these things because sometimes for lack of better ideas, science and engineering will proceed like this: 1) Take measurements of a system. 2) Plot the data. 3) Fit the data with a well-fitting mathematical function like the ones we've discussed. 4) Use the form of the curve fit to gain understanding about the system.
Looking at the air drag versus the viscous friction examples above is a perfect example. Clearly the two factors leading to air drag being proportional to speed squared can not both be present in viscous fluids. One was collisions, the other was due to passing between molecules more quickly to reach the next collision. In the case of viscous damping there is no meaningful space between the molecules. You can move an air molecule a long way before it hits its neighbor. On the other hand, if you move one long sugar molecule in honey, its neighbors are affected immediately since there are no gaps between. So the quadratic drag force becomes a linear viscous friction force because only the collision factor remains.
Of course there is a transition between the two cases (air drag and viscous damping) that must be gradual. The study of the details of fluid dynamics - which is really what that is - is a very complex one. I will leave it to your later education to fill in gaps about things like the Navier-Stokes equation, Reynolds numbers, turbulence and all the other topics that accompany studies of fluid dynamics.