50 CHAPTER 2. FUNCTIONS

Definition 2.1.13 Given two sets, X and Y, the Cartesian product of the two sets,written as X ×Y, is assumed to be a set described as follows.

X ×Y = {(x,y) : x ∈ X and y ∈ Y} .

R2 denotes the Cartesian product of R with R.

The notion of Cartesian product is just an abstraction of the concept of identifying apoint in the plane with an ordered pair of numbers.

Definition 2.1.14 Let f : D( f )→ R( f ) be a function. The graph of f consists ofthe set,{(x,y) : y = f (x) for x ∈ D( f )} .

Note that knowledge of the graph of a function is equivalent to knowledge of the func-tion. To find f (x) , simply observe the ordered pair which has x as its first position on leftand the value of y equals f (x) .

2.2 Graphs of Functions and RelationsRecall the notion of the Cartesian coordinate system you probably saw earlier. It involvedan x axis, a y axis, two lines which intersect each other at right angles and one identifiesa point by specifying a pair of numbers. For example, the number (2,3) involves going 2units to the right on the x axis and then 3 units directly up on a line perpendicular to the xaxis. For example, consider the following picture.

y

x

•(2,3)

Because of the simple correspondence between points in the plane and the coordinatesof a point in the plane, it is often the case that people are a little sloppy in referring to thesethings. Thus, it is common to see (x,y) referred to as a point in the plane. I will oftenindulge in this sloppiness. In terms of relations, if you graph the points as just described,you will have a way of visualizing the relation.

The reader has likely encountered the notion of graphing relations of the form y= 2x+3or y = x2 + 5. The meaning of such an expression in terms of defining a relation is asfollows. The relation determined by the equation y = 2x+ 3 means the set of all orderedpairs (x,y) which are related by this formula. Thus the relation can be written as{

(x,y) ∈ R2 : y = 2x+3}.

The relation determined by y = x2 + 5 is{(x,y) ∈ R2 : y = x2 +5

}. Note that these rela-

tions are also functions. For the first, you could let f (x) = 2x+ 3 and this would tell youa rule which tells what the function does to x. However, some relations are not functions.