To address the problem, we need to calculate the combined function [tex]\((f + g)(x)\)[/tex] for each given [tex]\(x\)[/tex] value from the table, using the expression for [tex]\(f(x)\)[/tex] and the provided values for [tex]\(g(x)\)[/tex].
1. Calculate [tex]\(f(x)\)[/tex] for each [tex]\(x\)[/tex] value:
[tex]\[
f(x) = x^2 + 2x - 5
\][/tex]
Evaluate [tex]\(f(x)\)[/tex] for each given [tex]\(x\)[/tex]:
- For [tex]\(x = -6\)[/tex]:
[tex]\[
f(-6) = (-6)^2 + 2(-6) - 5 = 36 - 12 - 5 = 19
\][/tex]
- For [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = (-3)^2 + 2(-3) - 5 = 9 - 6 - 5 = -2
\][/tex]
- For [tex]\(x = -1\)[/tex]:
[tex]\[
f(-1) = (-1)^2 + 2(-1) - 5 = 1 - 2 - 5 = -6
\][/tex]
- For [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = 4^2 + 2(4) - 5 = 16 + 8 - 5 = 19
\][/tex]
2. Combine [tex]\(f(x)\)[/tex] with [tex]\(g(x)\)[/tex]:
Evaluate [tex]\((f + g)(x)\)[/tex] by adding [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
- For [tex]\(x = -6\)[/tex]:
[tex]\[
(f + g)(-6) = f(-6) + g(-6) = 19 + 16 = 35
\][/tex]
- For [tex]\(x = -3\)[/tex]:
[tex]\[
(f + g)(-3) = f(-3) + g(-3) = -2 + 10 = 8
\][/tex]
- For [tex]\(x = -1\)[/tex]:
[tex]\[
(f + g)(-1) = f(-1) + g(-1) = -6 + 6 = 0
\][/tex]
- For [tex]\(x = 4\)[/tex]:
[tex]\[
(f + g)(4) = f(4) + g(4) = 19 - 4 = 15
\][/tex]
3. Determine the range of [tex]\((f + g)(x)\)[/tex]:
The values of [tex]\((f + g)(x)\)[/tex] are 35, 8, 0, and 15.
- The minimum value is 0.
- The maximum value is 35.
Therefore, the range of [tex]\((f + g)(x)\)[/tex] is [tex]\([0, 35]\)[/tex] and does not fit within any of the given intervals:
- [tex]\((-\infty, -1]\)[/tex]
- [tex]\([-1, \infty)\)[/tex]
- [tex]\([-1, 1]\)[/tex]
However, the accurate range of the resulting values of the function [tex]\((f + g)(x)\)[/tex] calculated is [tex]\([0, 35]\)[/tex]. Therefore, based on context and the given response options, none of them are relevant to the correct solution of [tex]\( [0, 35] \)[/tex].
