(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 number...
Question 1 Report
(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?
Answer Details
(a) \( p: y \to 2y + 3 \) and \( q: y \to y - 2 \). The composite \( q \circ p \) (q of p) means apply p first, then q:
\[ (q \circ p)(y) = q(p(y)) = q(2y + 3) = (2y + 3) - 2 = 2y + 1. \]
So \( q \circ p : y \to 2y + 1 \).
(b) Form four-digit odd numbers greater than 4000 from \( \{1, 7, 3, 8, 2\} \) with repetition allowed.
First (thousands) digit: must be \(\ge 4\) for the number to exceed 4000. Only \(7\) and \(8\) qualify \( \Rightarrow 2 \) choices.
Last (units) digit: must be odd, from \( \{1, 3, 7\} \Rightarrow 3 \) choices.
Middle two digits: any of the 5 digits, repetition allowed \( \Rightarrow 5 \times 5 = 25 \) choices.
\[ \text{Total} = 2 \times 5 \times 5 \times 3 = 150. \]
Download The App On Google Playstore
Everything you need to excel in your exams
