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Chapter 3
| Note: |
A function defines one variable in terms of another. The statement "y is a function of x" (denoted y = y(x)) means that y varies according to whatever value x takes on. A causal relationship is often implied (i.e. "x causes y"), but does not *necessarily* exist.
The number of hours you spend toiling away in Butler library may be a function of the number of classes you're taking. It's also likely to be a function of whether you have a computer in your dorm room, whether you have a printer in your dorm room, and whether your roommates are loud and obnoxious and won't let you study.
In turn, the number of classes you're taking may be a function of your major. It is also likely to be a function of how many extracurricular activities you participate in, how much sleep you need, and how crazy you are.
Knowing that one variable somehow changes with another is a starting point. Uncovering the specific nature of their relationship is another, and in science it is vital to understanding how systems behave. Such work can become a complicated business. Fortunately we have access to a language which, although it may have earned the undeserved reputation of being hopelessly complex, is actually the simplest, most efficient way to solve these problems. So let's do some translating:
If y(x) = 2x, that means that for every incremental increase in x, y increases by 2 increments. Y is twice the value of x, for every value x takes on.
If x = 5, then
y = 2 x 5 = 10.
If x = -0.2, then
y = 2 x (-0.2) = -0.4.
Y isn't always (or even usually) a function of just one variable. Y can vary
according to an infinite number of variables. Let's take the instance of two:
y = y(x, b) = which reads, "y is a function of x and b."
And
Y = number of days you've been alive
x = your age (i.e., number of birthdays passed)
b = number of days since your last birthday
so that
y(x) = 365x + x/4 + b.
The first "term" in the equation - 365x - says that each birthday corresponds to 365 days.
The second term - x/4 - takes into account that every fourth year you live includes one extra day - for leap year. (This term is actually a bit more complicated, but for our purposes it will do.)
The final term - b - takes care of the additional days since your last birthday.
So say it's September 7, 2004, and you will turn 18 on October 2nd. Then
x = 17
b = (366 - 25 = 341) (this year was a leap year, and there are 25 days
between Sept 7 and Oct 2.)
y = 365(17) + 17/4 + 341
= 6205 + 4.25 + 341 ~ 6550 days.