LECTURE 1
Review of Selected Topics
Functions
The Basic Concept
Let Xand Ybe sets of real numbers. A (numerical) function (from Xto Y)is intuitively described as a “rule” that
associates to each number xfrom the set Xone, and only one, number yin the set Y.
To speficy such a rule is to specify, for each value xin X, the corresponding value yin Y, i.e. to write down all
the ordered pairs
(x, y),
where xis in X, and yis the unique number in Yassociated to x.
Of course, if Xcontains infinitely many numbers, one cannot physically write down all such pairs. However, the
size of Xis a separate issue: even for an infinite X, one can meaningfully speak of specifying, in principle, for each x
in Xsome value yin Y. (Often, even with Xinfinite, this description can be given by a formula which, essentially,
tells how to compute y=f(x)from x. Most of the functions in this course will have infinite domain and a formulaic
description.)
Thus, describing a function amounts to describing–if only in principle–a collection of ordered pairs. One can
therefore give the following, precise, definition of a function.
Definition. A function from Xto Yis a set of ordered pairs
(x, y)
such that
•the first component of the pair, x, belongs to the set X,
•the second component of the pair, y, belongs to the set Y,
•every xfrom Xbelongs to exactly one pair, and
•no two distinct pairs share the same first component.
Each number xin Xis then called a value of the argument, and the corresponding yis the value of the function at
x.
Terminology and Notation
A function is frequently denoted by a symbol, e.g. the letter f, that can be conveniently reffered to later. If yis the
value of the function fat x, one writes
y=f(x)
The Vertical Line Test
The last bulleted condition in the above definition is often called the vertical line test, for the following reason. If f
is a function from Xto Y, the collection of all points
(x, y),where y=f(x),
in a coordinate plane is called the plot of fin that plane. The coordinate axis containing the values of Yis typically
drawn vertical. If the plot were to contain two points of the form (x, y1) and (x, y2), with y16=y2, then one would
be able to draw a “vertical” line through both points. Thus, no vertical line should be able to cross the plot of a
function more than once; this is the content of the vertical line test.
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