Read this section for an introduction to combinations of functions, then work through practice problems 1-9.
The graph of the greatest integer function has a break or jump at each integer value, but how many breaks can a function have? The next function illustrates just how broken or "holey" the graph of a function can be.
Define
Then , and since 3,
5/3 and –2/5 are all rational numbers. , and since and are all irrational numbers. These and some
other points are plotted in Fig. 16 .
In order to analyze the behavior of the following fact about rational and irrational numbers is useful.
Fact: "Every interval contains both rational and irrational numbers" or, equivalently, "If
and are real numbers and , then there is
The Fact tells us that between any two places where the (because is rational) there is a place where is 2 because there is an irrational number between any two distinct rational numbers. Similarly,
between any two places where (because is irrational) there is a place where because there is a rational number between any two distinct irrational numbers. The graph of
is impossible to actually draw since every two points on the graph are separated by a hole. This is also an example of a function which your computer or calculator can not graph because in general it can not determine whether
an input value of is irrational.
Example 7: Sketch the graph of
Solution: A sketch of the graph of is shown in Fig. 17 .
When is rational, the graph of looks like the "holey" horizontal line . When is irrational, the graph of looks like the "holey" line .