`f:xto` highest prime factor of x.
Find the range of f when the domain is `{12,13,14,15,16,17}`.

Answer

To find the range of the function \( f(x) \), where \( f(x) \) is defined as the highest prime factor of \( x \), we will evaluate this function for each element in the given domain \( \{12, 13, 14, 15, 16, 17\} \).

### Step-by-Step Solution:

1. **Evaluate \( f(12) \)**:
- The prime factorization of 12 is \( 2^2 \times 3^1 \).
- The highest prime factor is \( 3 \).
- Therefore, \( f(12) = 3 \).

2. **Evaluate \( f(13) \)**:
- The number 13 is a prime number.
- Thus, the highest prime factor of 13 is \( 13 \).
- Therefore, \( f(13) = 13 \).

3. **Evaluate \( f(14) \)**:
- The prime factorization of 14 is \( 2^1 \times 7^1 \).
- The highest prime factor is \( 7 \).
- Therefore, \( f(14) = 7 \).

4. **Evaluate \( f(15) \)**:
- The prime factorization of 15 is \( 3^1 \times 5^1 \).
- The highest prime factor is \( 5 \).
- Therefore, \( f(15) = 5 \).

5. **Evaluate \( f(16) \)**:
- The prime factorization of 16 is \( 2^4 \).
- The highest prime factor is \( 2 \).
- Therefore, \( f(16) = 2 \).

6. **Evaluate \( f(17) \)**:
- The number 17 is a prime number.
- Thus, the highest prime factor of 17 is \( 17 \).
- Therefore, \( f(17) = 17 \).

### Collecting the Results:
Now we have the following values:
- \( f(12) = 3 \)
- \( f(13) = 13 \)
- \( f(14) = 7 \)
- \( f(15) = 5 \)
- \( f(16) = 2 \)
- \( f(17) = 17 \)

### Finding the Range:
The range of the function \( f \) is the set of unique outputs we obtained:
- \( \{2, 3, 5, 7, 13, 17\} \)

### Final Answer:
Thus, the range of \( f \) when the domain is \( \{12, 13, 14, 15, 16, 17\} \) is:
\[ \{2, 3, 5, 7, 13, 17\} \]