Let `A = {9, 10 , 11 , 12 , 13}`and let `f : A ->N`be defined by f(n) = the highest prime factor of n. Find the range of f.

Explanation

To find the range of the function \( f : A \to \mathbb{N} \) defined by \( f(n) \) as the highest prime factor of \( n \) for the set \( A = \{9, 10, 11, 12, 13\} \), we will evaluate \( f(n) \) for each element in \( A \).

### Step-by-Step Solution:

1. **Evaluate \( f(9) \)**:
- The prime factorization of \( 9 \) is \( 3^2 \).
- The highest prime factor of \( 9 \) is \( 3 \).
- Thus, \( f(9) = 3 \).

2. **Evaluate \( f(10) \)**:
- The prime factorization of \( 10 \) is \( 2 \times 5 \).
- The highest prime factor of \( 10 \) is \( 5 \).
- Thus, \( f(10) = 5 \).

3. **Evaluate \( f(11) \)**:
- The number \( 11 \) is a prime number.
- The highest prime factor of \( 11 \) is \( 11 \) itself.
- Thus, \( f(11) = 11 \).

4. **Evaluate \( f(12) \)**:
- The prime factorization of \( 12 \) is \( 2^2 \times 3 \).
- The highest prime factor of \( 12 \) is \( 3 \).
- Thus, \( f(12) = 3 \).

5. **Evaluate \( f(13) \)**:
- The number \( 13 \) is also a prime number.
- The highest prime factor of \( 13 \) is \( 13 \) itself.
- Thus, \( f(13) = 13 \).

6. **Compile the results**:
- We have the following values for \( f(n) \):
- \( f(9) = 3 \)
- \( f(10) = 5 \)
- \( f(11) = 11 \)
- \( f(12) = 3 \)
- \( f(13) = 13 \)

7. **Determine the range**:
- The range of \( f \) is the set of unique values obtained from the function evaluations: \( \{3, 5, 11, 13\} \).

### Final Answer:
The range of the function \( f \) is \( \{3, 5, 11, 13\} \).