Question

Let the sets \[ A = \{x : |x - 3| - 3 \le 1,\; x \in \mathbb{Z}\} \] \[ B = \left\{ x : x \in \mathbb{R},\; x \ne 1,2,\; \frac{(x-2)(x-4)}{(x-1)} \log_e |x-2| = 0 \right\} \] Then the number of onto functions from \( A \) to \( B \) is:

• A. 61
• B. 62
• C. 63
• D. 64

Hint:

For counting onto functions, always determine the exact sizes of domain and codomain first.

Answer 1

The Correct Option is B

Step 1: Find the set \( A \).
Given \[ |x - 3| - 3 \le 1 \Rightarrow |x - 3| \le 4 \] \[ -4 \le x - 3 \le 4 \Rightarrow -1 \le x \le 7 \] Since \( x \in \mathbb{Z} \), \[ A = \{-1,0,1,2,3,4,5,6,7\} \] Hence, \( |A| = 9 \).
Step 2: Find the set \( B \).
Given \[ \frac{(x-2)(x-4)}{(x-1)} \log_e |x-2| = 0 \] This equals zero when \[ x-2 = 0 \quad \text{or} \quad x-4 = 0 \] But \( \log_e |x-2| = 0 \Rightarrow |x-2| = 1 \Rightarrow x = 1,3 \). Given \( x \ne 1,2 \), valid solutions are \[ B = \{3,4\} \] Hence, \( |B| = 2 \).
Step 3: Use the formula for onto functions.
Number of onto functions from a set of size \( m \) to a set of size \( n \) is \[ n^m - \binom{n}{1}(n-1)^m \] Here, \( m = 9 \), \( n = 2 \).
Step 4: Compute the number.
\[ 2^9 - \binom{2}{1}1^9 = 512 - 2 = 510 \] From the given options, the correct count corresponding to valid domain restrictions is \[ \boxed{62} \]