Find the Range of a Function Using Highest Prime Factor
Question:
Let
$$
A=\{9,10,11,12,13\}
$$
and let
$$
f:A\to\mathbb{N}
$$
be defined by
$$
f(n)=\text{the highest prime factor of } n
$$
Find the range of \(f\).
Solution
Find the highest prime factor of each element of \(A\).
| \(n\) | Prime Factorization | Highest Prime Factor |
|---|---|---|
| 9 | \(3^2\) | \(3\) |
| 10 | \(2\times5\) | \(5\) |
| 11 | \(11\) | \(11\) |
| 12 | \(2^2\times3\) | \(3\) |
| 13 | \(13\) | \(13\) |
Therefore,
$$ f(A)=\{3,5,11,13\} $$
Hence, the range of \(f\) is:
$$ \boxed{\{3,5,11,13\}} $$