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) and then 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?

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