Graphs of functions may be sketched in the usual way.
Sketch the graph of the function f given by:
State the domain and range of f.
Any set B which contains the range of a function (ie. ) is called a co-domain of the f. The notation (read: f from A to B) means simply f is a function whose domain is in A (ie. ) and whose range is in B (perhaps equal to B).
In practice mathematicians still refer to expressions in x as functions without mentioning any domain. When this is the case it is an unwritten law that one may assume the domain as the largest subset of the reals for which the expression is meaningful.
What are the intended domains when we reference the functions
First a word about numbers used in Mathematics.
– the natural numbers.
– the integers.
– the rational numbers (really the ratio numbers). The decimal representation of each rational is either terminating or recurring. Also block recurring decimals are rational. We can represent each rational number as a point on a number line with 0 and 1 marked.
But there are points on the line which are not the representation of a rational number. For example is not rational (proved in the analysis course) but is on the line.
Now it can be shown (although it is rather hard) that each point of the line corresponds to some decimal that may or may not be block recurring.
– the real numbers. The most elementary way of thinking of the reals is as decimals or as all the points on the line. In fact we often speak of the real line. Decimals which are not block recurring are said to be irrational.
For practical purposes the rational numbers are enough because any irrational can be approximated as closely as we like by a rational. For most mathematical purposes however it is convenient to have the irrationals as well.
– the complex numbers. These are thought of as numbers of the form where a and b are real.