self-inverse and periodic functions

some functions, if repeatedly used (i.e. iterated) return to a start value and it can be interesting to explore these - by operating with fractions preferably, or using a calculator

it can be demanding to firmly establish how many steps it takes to return to a start value (the period) algebraically

the 'usual' self-inverse functions are of a form similar to:
f(x) = 10 - x and
f(x) = 12/x

what happens if you work out 10 - 3 and then subtract the result from 10?
leading to an appreciation that 10 - (10 - n) is n

however, there are many more examples of self-inverse rational functions
here are a few:

etc.
etc.
there is a simple enough rule for such functions to be self-inverse

but it's not that easy to justify...

for practice in dividing fractions and working with directed numbers, self-inverse functions provide an interesting context:

• put a number ('seed' value) into one of the functions (above) 
• feed the output (y-value) back into the function

what happens?
try this with several input (start) numbers...


why is the process problematic if the denominator is = 0?
other than for a problematic x value, does the number always seem to return to the start?
what if x is a fraction, or a negative number, or a negative fraction?

what happens with these functions?