Question

The number of functions f , from the set
$A = {x∈N: x^2-10x+9≤0}$to the set $B = {n62:n∈N}$
such that
$f(x)≤(x-3)^2+1$, for every $x∈A,$
is ______.

Answer 1

A = {x∈N, x2-10x+9≤0}
= {1,2,3,...,9}
B = {1,4,9,16,....}
f(x)≤(x-3)2+1
f(1)≤5, f(2)≤2,....f(9)≤37
x = 1 has 2 choices
x = 2 has 1 choice
x = 3 has 1 choice
x = 4 has 1 choice
x = 5 has 2 choices
x = 6 has 3 choices
x = 7 has 4 choices
x = 8 has 5 choices
x = 9 has 6 choices
∴ Total functions = 2 × 1 × 1 × 1 × 2 × 3 × 4 × 5 × 6 = 1440