Find the Range of a Function Using Highest Prime Factor
Question:
Let
$$
A=\{12,13,14,15,16,17\}
$$
and
$$
f:A\to \mathbb{Z}
$$
be a function given by
$$
f(x)=\text{highest prime factor of }x
$$
Find the range of \(f\).
Solution
Find the highest prime factor of each element of \(A\).
| \(x\) | Prime Factorization | Highest Prime Factor |
|---|---|---|
| 12 | \(2^2\times3\) | \(3\) |
| 13 | \(13\) | \(13\) |
| 14 | \(2\times7\) | \(7\) |
| 15 | \(3\times5\) | \(5\) |
| 16 | \(2^4\) | \(2\) |
| 17 | \(17\) | \(17\) |
Therefore,
$$ f(A)=\{2,3,5,7,13,17\} $$
Hence, the range of \(f\) is:
$$ \boxed{\{2,3,5,7,13,17\}} $$