The number of functions f , from the set A = \ x ∈ N: x ^ 2 - 10x +...

For set $A$

\left({x}^{2} - 10x + 9\right) \le 0 ⇒ \left(x - 1\right)\left(x - 9\right) \le 0$</p><p>$⇒ x \in \left[1, 9\right]$</p><p>i.e.$A = \left\{1, 2, 3, 4, 5, 6, 7, 8, 9\right\}

Given $B = \left\{{n}^{2}: n \in N\right\}$

Now $f\left(x\right) \le {\left(x - 3\right)}^{2} + 1$

For $x = 1: f\left(1\right) \le 5 ⇒ B \in \left\{{1}^{2}, {2}^{2}\right\}$

For $x = 2: f\left(2\right)$ \le 2 ⇒ B \in \left\{{1}^{2}\right\}$</p><p>For$x = 3: f\left(3\right) \le 1 ⇒ B \in \left\{{1}^{2}\right\}

For $x = 4: f\left(4\right) \le 2 ⇒ B \in \left\{{1}^{2}\right\}</p><p>For$x = 5: f\left(5\right)$\le 5 ⇒ B \in$\left\{{1}^{2}, {2}^{2}\right\}$</p><p>For$x = 6: f\left(6\right)$\le 10 ⇒ B \in \left\{{1}^{2}, {2}^{2}, {3}^{2}\right\}$

For $x = 7: f\left(7\right) \le 17 ⇒ B \in \left\{{1}^{2}, {2}^{2}, {3}^{2}, {4}^{2}\right\}</p><p>For$x = 8: f\left(8\right) \le 26 ⇒ B \in \left\{{1}^{2}, {2}^{2}, {3}^{2}, {4}^{2}, {5}^{2}\right\}

For $x = 9: f\left(9\right)$ \le 37 ⇒ B \in \left\{{1}^{2}, {2}^{2}, {3}^{2}, {4}^{2}, {5}^{2}, {6}^{2}\right\}$</p><p>Total number of such function</p><p>$= 2 \times 1 \times 1 \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 2\left(720\right) = 1440