Let A = {9, 10, 11, 12, 13} and let f : A → Z be a function given by f(n) = the highest prime factor of n. Find the range of f.
Given A = {9, 10, 11, 12, 13}
f : A → Z such that f(n) = the highest prime factor of n.
A is the domain of the function f. Hence, the range is the set of elements f(n) for all n ∈ A.
We have f(9) = highest prime factor of 9
The prime factorization of 9 = 32
Thus, the highest prime factor of 9 is 3.
∴ f(9) = 3
We have f(10) = highest prime factor of 10
The prime factorization of 10 = 2 × 5
Thus, the highest prime factor of 10 is 5.
∴ f(10) = 5
We have f(11) = highest prime factor of 11
We know 11 is a prime number.
∴ f(11) = 11
We have f(12) = highest prime factor of 12
The prime factorization of 12 = 22 × 3
Thus, the highest prime factor of 12 is 3.
∴ f(12) = 3
We have f(13) = highest prime factor of 13
We know 13 is a prime number.
∴ f(13) = 13
Thus, the range of f is {3, 5, 11, 13}.