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1 Functions
In this Chapter we will cover various aspects of functions. We will look at the definition of
a function, the domain and range of a function, what we mean by specifying the domain
of a function and absolute value function.
1.1 What is a function?
1.1.1 Definition of a function
A function f from a set of elements X to a set of elements Y is a rule that
assigns to each element x in X exactly one element y in Y .
One way to demonstrate the meaning of this definition is by using arrow diagrams.
f : X → Y is a function. Every element
in X has associated with it exactly one
element of Y .
g : X → Y is not a function. The ele-
ment 1 in set X is assigned two elements,
5 and 6 in set Y .
A function can also be described as a set of ordered pairs (x, y) such that for any x-value in
the set, there is only one y-value. This means that there cannot be any repeated x-values
with different y-values.
The examples above can be described by the following sets of ordered pairs.
F={(1,5),(3,3),(2,3),(4,2)} is a func-
tion.
G={(1,5),(4,2),(2,3),(3,3),(1,6)} is not
a function.
The definition we have given is a general one. While in the examples we have used numbers
as elements of X and Y , there is no reason why this must be so. However, in these notes
we will only consider functions where X and Y are subsets of the real numbers.
In this setting, we often describe a function using the rule, y = f (x), and create a graph
of that function by plotting the ordered pairs (x, f(x)) on the Cartesian Plane. This
graphical representation allows us to use a test to decide whether or not we have the
graph of a function: The Vertical Line Test.