Function Notation
When we speak of "function notation" we mean writing a function
in the form such as
f(x) = 2x + 1
We should note, however, that a function is a special type of a relation,
and that a relation is a set of ordered pairs.
The function notation example given above is actually an abbreviation
for f = {(x,y) | y = 2x + 1 }
In this longer form it is clear that f is the name of a set of
ordered pairs (that is, it is the name of a relation), with the extra restriction that
for any particular ordered pair in the function, the second coordinate is determined to be one more than twice the
first coordinate. We will often abbreviate this long form by just writing
y = 2x + 1
and, at times we will use the "function notation"
f(x) = 2x + 1
We will complicate matters even further by writing, at times, y=f(x). (The complexity of
this becomes evident when we reverse the "equality" and write f(x)=y and then try to
explain that form the same way that we explain f(x)=2x+1.) And, we even go so far
as to write the ordered pair (x,f(x)) when we mean the ordered pair
(x,y) where y = f(x) and f is really
the name of a function.
When we write f(x) = 2x + 1 we are not writing an "equality" as much as we are
writing a "definition". Really, we are
saying that f is a function defined so that the output of the function (the range element)
is determined to be equal to one more than twice the input to the function (the domain element).
PRECALCULUS: College Algebra and Trigonometry
© 2000 Dennis Bila, James Egan, Roger Palay