Input numbers have been changed by a rule to get output numbers which of these could be the rule for changing the input numbers to the output numbers? input: 3,6,9. output: 4,5,6.

Changes in output are evenly spaced for evenly spaced changes in input, so the rule is linear. Changes in output are 1 for changes in input of 3, so the rule involves a factor of 1/3. One possible rule is

... output = (1/3)·input + 3

If y and x are used to represent output and input, respectively, this is

... y = (1/3)x + 3

ok thats very interesting