Problem: gerri made a function machine. find the rule for gerri's function machine.
-1 goes in -1 comes out. what functions went on?

There are an infinite number of possibilities, and not enough information
to decide which possibility is really the one inside the function machine.

Here are a few.  Each of these gives the result that you described,
and there are an infinite number of others:

f(x) = x

f(x) = 2x + 1

f(x) = 10x + 9

f(x) = x² - 2

f(x) = 7x² - 8

f(x) = 31x³ + 30

f(x) = log( |x| ) - 1

f(x) = ln( |x| ) - 1

f(x) = x tan(45°)
.
.
etc.