Explanation:
f(x) = 12 × 4^x = 3×4 × 4^x = 3 × 4^(x + 1)
on day 0 (x = 0) the new participants are 3×4 = 12, as each of the 3 founders is sending the message out to 4 friends. so, 12 contacted friends as new participants.
on day 1 (x = 1), the new participants are 3×4×4 = 48, as each of the 12 new participants of day 0 are sending out 4 new messages.
on day 2 (x = 2), the new participants are 3×4×4×4 = 192, as each of the 48 new participants of day 1 are sending out 4 new messages.
and so on.
every new day the number of new participants of the previous day sends out 4 new messages, so, the number of the new participants of the new day is 4 times the number of new participants of the previous day.
so, the result of the previous day is multiplied by 4. and that is multiplied by 4. and that again. and so on.
but the starting value is 12. therefore, we need the 3×4 at the beginning, and we keep it, of course, as the foundation for all the future multiplications by 4.