Consider two sets A = { x : | | x - 3 | - 3 | 1 and B = { x - { 1 , 2 } : ( x - 2 ) ( x - 4 ) x - 1 ? log e ( | x - 2 | ) = 0 } . Then the number of onto functions f : A B is equal to. [2026]

Question

Consider two sets $A = {x \in ℤ : | (| x - 3 | - 3) | \leq 1$ and

$B = {x \in ℝ - {1, 2} : \frac{(x - 2) (x - 4)}{x - 1} \log_{e} (| x - 2 |) = 0} .$

Then the number of onto functions $f : A \to B$ is equal to. [2026]

Smart Achievers · Accepted Answer

(1)

$A : | | x - 3 | - 3 | \leq 1$

$\Rightarrow - 1 \leq | x - 3 | - 3 \leq 1$

$\Rightarrow 2 \leq$ $| x - 3 |$ $\leq 4$

$\Rightarrow 2 \leq (x - 3) \leq 4 or - 4 \leq (x - 3) \leq - 2$

$\Rightarrow 5 \leq x \leq 7 or - 1 \leq x \leq 1$

$A = {- 1, 0, 1, 5, 6, 7}$

$B \Rightarrow x = 4, | x - 2 | = 1 \Rightarrow x = 3 or 1 (reject)$

$\Rightarrow B = {3, 4}$

$Number of onto functions from A to B = 2^{6} - 2 = 62$

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