The number of functions $f$, from the set $\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}$ to the set $\mathrm{B}=\left\{\mathrm{n}^{2}: \mathrm{n} \in \mathbf{N}\right\}$ such that $f(x) \leq(x-3)^{2}+1$, for every $x \in \mathrm{A}$, is ___________.
Answer (integer)
1440
Solution
<p>$$A = \left\{ {\matrix{
{x \in N,} & {{x^2} - 10x + 9 \le 0} \cr
} } \right\}$$</p>
<p>$= \{ 1,2,3,\,....,\,9\}$</p>
<p>$B = \{ 1,4,9,16,\,.....\}$</p>
<p>$f(x) \le {(x - 3)^2} + 1$</p>
<p>$f(1) \le 5,\,f(2) \le 2,\,\,..........\,f(9) \le 37$</p>
<p>$x = 1$ has 2 choices</p>
<p>$x = 2$ has 1 choice</p>
<p>$x = 3$ has 1 choice</p>
<p>$x = 4$ has 1 choice</p>
<p>$x = 5$ has 2 choices</p>
<p>$x = 6$ has 3 choices</p>
<p>$x = 7$ has 4 choices</p>
<p>$x = 8$ has 5 choices</p>
<p>$x = 9$ has 6 choices</p>
<p>$\therefore$ Total functions = $2\times1\times1\times1\times2\times3\times4\times5\times6=1440$</p>
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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