find f(x) = x(mod 5).  Namely, find the remainder of x after division by 5.

Method:  Delete 1's keeping count of how many until 5 have been deleted.  
         Repeat this until you run out of 1's.  Then from the count you
         know how many to return to the tape.

Final state = {9}.
-------------------------------------------------

Count 5 1's.  state number tells how many have been counted

(01 1r   State 0:  have counted a multiple of 5 1's
(11 2r
(21 3r
(31 4r
(41 0r

---------------

Ran out of 1's.  Now put back the right amount.

(0  5l     No 1's to put back.  State 5:  go back to front of tape.
(1 10r
(2 11r
(3 12r
(4 13r    four 1's to write back.

---------------

Go to front of string....
(5115l
(5  9r