(a) Two functions p and q are defined on the set of real numbers, R, by p : y \(\to\) 2y +3 and q : y -> y - 2. Find QOP
(b) How many four digits odd numbers greater than 4000 can be formed from 1,7,3,8,2 if repetition is allowed?
(a) To find QOP: we need to apply the functions in the correct order. QOP means we apply Q first, then O, then P.
Q: y → y - 2, so QOP: y → 2(y - 2) + 3 = 2y - 1. Therefore, QOP: y → 2y - 1.
(b) To form a four-digit odd number greater than 4000: the leftmost digit must be 7. We can choose any of the remaining digits for the thousands place, giving us 4 choices. For the hundreds, tens, and ones places, we can choose any of the five digits (1, 3, 7, 8, 2) with repetition allowed, giving us 5 choices for each of those places.
Therefore, the total number of four-digit odd numbers greater than 4000 that can be formed with these digits is:
4 (choices for the thousands place) × 5 (choices for the hundreds place) × 5 (choices for the tens place) × 5 (choices for the ones place) = 500.
So, there are 500 four-digit odd numbers greater than 4000 that can be formed from 1, 7, 3, 8, 2 if repetition is allowed.