Class Content I >Modeling with mathematics > Math recap

2.2.4

Functions

One of the key mathematical ideas in the use of math in science is the idea of a function. Functions simply mean that one thing depends on another. The simplest math example is a single independent variable (call it x) and a single dependent variable (call it y).  The fact that y takes on values that are determined by the value that x has following some rule is written:

y = f (x )

We write "y" when we are focusing on the value that is produced and we write f (x ) when we are focusing on what we do to x  in order to get the value that we are going to put into y.

Note that in order to have a function, we don't have to have a simple algebraic expression that works for all values of x — and it doesn't even have to be defined for all values of x. For example, we could have a function that assigns the value of "1" to every decimal that has an even digit in its 12th place and "0" to every other one. This is still a function — though not a very useful one.  We could also have functions that are defined differently in different regions.  See for example our problem on the propagation of a triangular wave. (Note this problem also writes the pulse in terms of dimensionless ratios so as to get the units right and still reduce to something that looks like "math-without-units".)

Note also that in scientific use of math we almost never have a result that depends on a single input. We often have to pay attention to lots of variables — even some that we think of as constants. And what is a "dependent" or "independent" variable might depend on what choice we make of what experiment to set up or what we can control.

Functional Dependence: It's the relationship that matters

The critical part about a function is not just that two variables are related but how they are related.  Often in biological applications of physical principles there will be competing effects that vary in different ways.  The specific way that these effects vary can be critical. For example, in our problems about scaling (e.g., the worm) one effect depends on an object's surface while another effect depends on the object's volume. If the object grows isometrically (all dimensions increase together) then the surface grows with the square of the length and the volume with the cube of the length. This may constrain the way the object can grow and can have profound implications for evolution.

Here are some of the kinds of dependencies we will be using in this class:

• Linear:              f (x ) = x * (stuff that does not depend on x )
• Quadratic         f (x ) = x² * (stuff that does not depend on x )
• Cubic               f (x ) = x³ * (stuff that does not depend on x )
• Inverse linear   f (x ) = (1/x ) * (stuff that does not depend on x )
• Inverse square f (x ) = (1/x²) * (stuff that does not depend on x )
• Exponential     f (x ) = e^x * (stuff that does not depend on x )

Note that the "stuff that does not depend on x" can be very messy and include lots of other variables and parameters! It's important that you learn to be able to choose to see some of the variables and parameters as "just constants" when you are considering the variation of a particular variable or parameter — and be able to switch which one you are focusing on!

Joe Redish 8/14/11