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 £© Functions And Their Graphs
Since functions are the basic building blocks out of which mathematicians construct models
of the physical world, it is essential that any student of mathematics have a firm grasp of
the concept. In particular, one must be careful to distinguish between a given function and
a notational or graphical representation for it. A function is a type of relationship, a mental
concept that cannot be seen or touched. Although pictures and symbolic representations
of a function are extremely important in understanding its behavior, the student must
always keep in mind the distinction between the function itself and its representations.
Modern methods for giving a formal definition of a function, developed in the latter
part of the 19th century, are based on set-theoretic ideas. We will not go into the details
necessary to make such a precise definition, but rather aim at an intuitive understanding
of the basic concept. For us, a function is a special type of relationship between two
quantities. We often think of this relationship to be one of dependence. That is, if the
value of one quantity, say y, is determined by the value of another quantity, say x, then we
say that y is a function of x. For example, if x represents the height from which a certain
rock is dropped and y represents the velocity with which the rock strikes the ground, then
the value of y will depend on the value of x and we say that velocity y is a function of height
x. Note here that if y is the terminal velocity of the object, then there are many different
values of x which yield the same value of y, namely, any value of x which gives the object
sufficient time to reach its terminal velocity before striking the ground. On the other hand,
for a given value of x, there is only one related value of y. It is this latter property that
makes the relationship between height