Functions
Next to sets, the most important types of object in
mathematic are functions. The functions you see most often in algebra and
calculus courses involve real numbers and have formulas such as x2,
log2(x), and sin x. Each of these expressions specifies a real
number that is associated with a real
number x; in the second case. It is necessary to say a positive real number x,
since log2(x) makes sense only if x>0. More generally, a fuction
is f, and the domain and codomain of f
are A and B, respectively, the element of B that f associates with an element x
of A written f(x). We Write
F:
A--> B
To indicate that f is a function with domain A and codomain
B. In informal terminology, we say f assigns an element of B to each element of
A; or f maps an element x of A to the element f(x) of B. A slightly more
preciseway of describing a function eliminates the need for such undefined term as “assigns”,”associates”, or “map”; we
shall discuss it a little later. But for most or our purposes, the informal
terminology is perfectly adequate.
Sometimes there seems to be a certain arbitrariness about
the codomain of a function. For example, consider the function f, whose domain
is the set of real number,given by the formula f(x) = x2. Let R be
the set of real numbers, and let R+ be the set of nonnegative real
number. We might say f: R à
R, since for every real number x, x2 is a nonnegativereal number. W might
also say f : RàR+,
since for every real number x, x2 is a nonnegative real number. To confuse
the matter more, if we were only
interested in considering f(x) for nonnegative value of x, we might say
f : R+ àR,
or f :R+ àR+.
to be precise, these are really four different functions, even though their values
for given x are all described by the same formula. To specify a function, we
must say what its domain and codomain are, and we must state the rule that
determines the value f(x) that assigned to each
x in the domain. This is law that is often broken : people speak of “the
function f(x)=x2”, or “ “the function log2(x).” One
common convention is to assume that the domain is the largest set for which the
formula makes sence for x2, R; for log2(x), {x ͼ R|x>0}. However, this still leaves the
codomain up in the air. In many cases, it may not matter, but in defining a
function we shall try to specify both sets.
