One Way We Write Functions

During the Summer A To Z I talked a bit about functions. Mathematically we see these as a collection of three things: a set of things which we call the domain, a set of things which we call the range, and a rule that matches things in the domain to something in the range. The domain and the range can be the same set, or they can be different ones. The definition is quite flexible. What I want to talk about here is how to write them down.

We can describe each of these sets in words, and often will when speaking or when describing a line of argument. But when we want to work, we start using shorthand names, often single letters. For the sets of the domain and range these are usually capital letters. I haven’t noticed much of a preference for which letters to use. D for domain and R for range have a hard-to-resist logic if we don’t really care what the sets are.

There are some sets that are used as domains or ranges a lot, and those have common shorthands. The set of real numbers is often written as R — bold, in print, or written with a double vertical stroke on the R if you’re doing this by hand. The set of whole numbers, integers, gets written as I (for integer) or J (again for integer; the letters I and J weren’t perceived as truly different things until recently) or Z (for Zahlen, German for “counting number”). There are a lot of others and don’t worry about them.

The rules for a function are generally described by a lowercase letter. It’s most commonly f, with g and h pressed into service if f won’t do. Subscrips are common also: f₁, f₂, f_j, f_n, and so on. Again, any name is allowed, as long as you’re consistent about it. But f and g and h are used as “names of functions” so often that it’s what the reader will expect they mean even without being told.

One common shorthand for saying that a function named “f” has the domain “D” and the range “R” is to use an arrow. Write out “f: D –> R”. The function name comes first, before the colon; then the domain, and an arrow, and the range. There are other notations but this is the one I see most often. This is often read aloud as “f maps D into R”. The activity of the verb “map” — well, it’s kind of action-y — suggests motion to my mind. Functions are commonly used to describe how a system changes over time. This seems mnemonic to me, as arrows suggest flow and motion. We often use the language of flowing things even for problems that don’t have anything to do with moving objects or any sense of time.

There’s another part of function-defining that has to be done, though. Most often we’re interested in domains and ranges that are both numbers, or at least collections of numbers. And we want to describe matching something in the domain with something in the range based on a formula. If “x” is a number in the domain then, say, “x² – 4x + 4” is the corresponding number in the range.

One way to write down this rule is the way we get in introductory algebra class, and to write something like “f(x) = x² – 4x + 4”. The “x” is, here, a dummy variable. We will never care about pinning it down to any particular number. If we write “f(3)” we mean to evaluate whatever’s on the right hand of the equals sign, using 3, the thing in parentheses, wherever “x” appears in the rule definition. In this case that would be the number 3² – 4*3 + 4 which it happens is 1. If we write “f(1 – t)” we would evaluate “(1 – t)² – 4(1 – t) + 4” which we might want to leave as is, or might want to simplify somehow. It depends what we’re up to.

But we can also use an arrow notation, and write the same rule as “f: x –> x² – 4x + 4”. My feeling is this notation makes it clearer that the definition isn’t itself something to solve, and that the definition doesn’t care what value x is. It should suggest how we can substitute anything for x and should do so throughout the expression to the right of the arrow.

Wikipedia asserts that when writing the rule this way there should be a vertical stroke on the left side of the arrow. This is probably a good rule, since “f: D –> R” and “f: x –> x² – 4x + 4” are talking about different things. I’m not sure the rule is consistently followed, though. I suspect that in most contexts it’s clear what is meant.