Recursive functions like this can be represented graphically by considering the line y=x and all the points of f(x) as in the graph below:

(1) Take your first point from f(x). (2) Draw a line horizontally to meet the line y=x. (3) Draw a line vertically to meet a point on the line f(x). There will never be more than one point with the same x value as f(x) because f(x) has a unique value. (4) Repeat steps two and three. If we take a look at the graphs of y=x and y=f(x) we can see that only a few points of f(x) lie above the line y=x. For these points alone f(x) > x as for all other points step 2 will draw a line to the left i.e decreasing in value.

We can see the same thing analytically. The terms increase if and only if 2a + 3b > 10a + b, that is iff b > 4a. As b is a single digit number the only terms that are followed by larger terms are 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17, 18, 19 and 29 as seen on the graph.

Large terms reduce by one fifth in order of magnitude, and once the sequence reaches 2 digit numbers the largest value is given by a=b=9 which maps to 45. For numbers less than 45 2a+3b < 33. As there are a finite number of values the sequence can take, values are repeated and cycles occur.

There are three points that lie on the line y=x. These are 0, 14 and 28 for which values x=f(x). Alternatively, we see that for a fixed point 10a + b = 2a + 3b so 4a=b. Hence the only fixed points are 0, 14 and 28.

There are eight distinct loops (five 6-cycles, one 2-cycle and three fixed points) which are:
[0, 0]
[2, 6, 18, 26, 22, 10, 2]
[4, 12, 8, 24, 16, 20, 4]
[5, 15,17, 23, 13, 11, 5]
[7, 21, 7]
[9, 27, 25, 19, 29, 31, 9]
[14, 14]
[28, 28]

Other terms which are not in the cycles but lead into the cycles are:
1,3, ® 9 etc.
30 ® 6 etc.
32 ® 12 etc.
33 ® 15 etc.

We have proved that all higher numbers reduce to 33 or less so sequence with all other starting points enter one of the listed cycles or end up at a fixed point.

Why are all sequences all odd or all even?
even= 0 (mod 2)
odd=1 (mod 2)

When deciding if a number is odd or even we only need to look at the last digit, in this question defined as 'b'. The only operation on b is b® 3b and 3×0 º 0mod 2 so it is even and 3×1 = 3 º 1 mod 2 so it is odd. Therefore f(x) º x mod 2.

The sequences are all odd or all even because

2a + 3b º 10a + b     mod 2.