To determine the range of the function \( g(x) = x^2 + 3x - 5 \) where the domain \( P = \{x: x \) is a factor of 6\}, we need to evaluate \( g(x) \) for each factor of 6. Factors of 6 are 1, 2, 3, and 6. Let's calculate \( g(x) \) for each of these values.
1. For \( x = 1 \):
[tex]\[
g(1) = 1^2 + 3(1) - 5 = 1 + 3 - 5 = -1
\][/tex]
2. For \( x = 2 \):
[tex]\[
g(2) = 2^2 + 3(2) - 5 = 4 + 6 - 5 = 5
\][/tex]
3. For \( x = 3 \):
[tex]\[
g(3) = 3^2 + 3(3) - 5 = 9 + 9 - 5 = 13
\][/tex]
4. For \( x = 6 \):
[tex]\[
g(6) = 6^2 + 3(6) - 5 = 36 + 18 - 5 = 49
\][/tex]
Thus, the corresponding values of \( g(x) \) for \( x = 1, 2, 3, 6 \) are \(-1, 5, 13, \) and \( 49 \), respectively. Therefore, the range of \( g(x) \) is the set of these values.
From this, we determine that the range is:
[tex]\[
\{-1, 5, 13, 49\}
\][/tex]
So, the correct answer is:
[tex]\[ \boxed{\{-1, 5, 13, 49\}} \][/tex]
