Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

0.4. Functions 43
We will also be interested in functions with domain N. Here two-line
notation is no good, but describing the function algebraically is often
possible. Even tables are a little awkward, since they do not describe the
function completely. For example, consider the function f : N → N given
by the table below.
x 0 1 2 3 4 5 . . .
f (x) 0 1 4 9 16 25 . . .
Have I given you enough entries for you to be able to determine f (6)?
You might guess that f (6) 36, but there is no way for you to know this for
sure. Maybe I am being a jerk and intended f (6) 42. In fact, for every
natural number n, there is a function that agrees with the table above, but
for which f (6) n.
Okay, suppose I really did mean for f (6) 36, and in fact, for the rule
that you think is governing the function to actually be the rule. Then
I should say what that rule is. f (n) n 2 . Now there is no confusion
possible.
Giving an explicit formula that calculates the image of any element in
the domain is a great way to describe a function. We will say that these
explicit rules are closed formulas for the function.
There is another very useful way to describe functions whose domain
is N, that rely specifically on the structure of the natural numbers. We can
define a function recursively!
Example 0.4.4
Consider the function f : N → N given by f (0) 0 and f (n + 1)
f (n) + 2n + 1. Find f (6).
Solution. The rule says that f (6) f (5) + 11 (we are using 6 n + 1
so n 5). We don’t know what f (5) is though. Well, we know that
f (5) f (4) + 9. So we need to compute f (4), which will require
knowing f (3), which will require f (2),. . . will it ever end?
Yes! In fact, this process will always end because we have N as
our domain, so there is a least element. And we gave the value of
f (0) explicitly, so we are good. In fact, we might decide to work up
to f (6) instead of working down from f (6):
f (1) f (0) + 1 0 + 1 1
f (2) f (1) + 3 1 + 3 4
f (3) f (2) + 5 4 + 5 9
f (4) f (3) + 7 9 + 7 16
f (5) f (4) + 9 16 + 9 25