To solve for [tex]\((f - g)(x)\)[/tex] given the functions [tex]\(f(x) = x^2 + 1\)[/tex] and [tex]\(g(x) = 5 - x\)[/tex], we should first derive the function [tex]\((f - g)(x)\)[/tex].
Step-by-Step Solution:
1. Define the Functions:
[tex]\[ f(x) = x^2 + 1 \][/tex]
[tex]\[ g(x) = 5 - x \][/tex]
2. Compute [tex]\((f - g)(x)\)[/tex]:
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
Substitute the given functions:
[tex]\[
(f - g)(x) = (x^2 + 1) - (5 - x)
\][/tex]
3. Simplify the Expression:
[tex]\[
(f - g)(x) = x^2 + 1 - 5 + x
\][/tex]
[tex]\[
(f - g)(x) = x^2 + x - 4
\][/tex]
Next, we need to evaluate [tex]\((f - g)(x)\)[/tex] for various values. Let's initially evaluate it for [tex]\(x\)[/tex] in the range of integers from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex]:
[tex]\[
\begin{aligned}
& \text{For } x = -10: & (f - g)(-10) &= (-10)^2 + (-10) - 4 = 100 - 10 - 4 = 86 \\
& \text{For } x = -9: & (f - g)(-9) &= (-9)^2 + (-9) - 4 = 81 - 9 - 4 = 68 \\
& \text{For } x = -8: & (f - g)(-8) &= (-8)^2 + (-8) - 4 = 64 - 8 - 4 = 52 \\
& \text{For } x = -7: & (f - g)(-7) &= (-7)^2 + (-7) - 4 = 49 - 7 - 4 = 38 \\
& \text{For } x = -6: & (f - g)(-6) &= (-6)^2 + (-6) - 4 = 36 - 6 - 4 = 26 \\
& \text{For } x = -5: & (f - g)(-5) &= (-5)^2 + (-5) - 4 = 25 - 5 - 4 = 16 \\
& \text{For } x = -4: & (f - g)(-4) &= (-4)^2 + (-4) - 4 = 16 - 4 - 4 = 8 \\
& \text{For } x = -3: & (f - g)(-3) &= (-3)^2 + (-3) - 4 = 9 - 3 - 4 = 2 \\
& \text{For } x = -2: & (f - g)(-2) &= (-2)^2 + (-2) - 4 = 4 - 2 - 4 = -2 \\
& \text{For } x = -1: & (f - g)(-1) &= (-1)^2 + (-1) - 4 = 1 - 1 - 4 = -4 \\
& \text{For } x = 0: & (f - g)(0) &= (0)^2 + (0) - 4 = 0 + 0 - 4 = -4 \\
& \text{For } x = 1: & (f - g)(1) &= (1)^2 + (1) - 4 = 1 + 1 - 4 = -2 \\
& \text{For } x = 2: & (f - g)(2) &= (2)^2 + (2) - 4 = 4 + 2 - 4 = 2 \\
& \text{For } x = 3: & (f - g)(3) &= (3)^2 + (3) - 4 = 9 + 3 - 4 = 8 \\
& \text{For } x = 4: & (f - g)(4) &= (4)^2 + (4) - 4 = 16 + 4 - 4 = 16 \\
& \text{For } x = 5: & (f - g)(5) &= (5)^2 + (5) - 4 = 25 + 5 - 4 = 26 \\
& \text{For } x = 6: & (f - g)(6) &= (6)^2 + (6) - 4 = 36 + 6 - 4 = 38 \\
& \text{For } x = 7: & (f - g)(7) &= (7)^2 + (7) - 4 = 49 + 7 - 4 = 52 \\
& \text{For } x = 8: & (f - g)(8) &= (8)^2 + (8) - 4 = 64 + 8 - 4 = 68 \\
& \text{For } x = 9: & (f - g)(9) &= (9)^2 + (9) - 4 = 81 + 9 - 4 = 86 \\
& \text{For } x = 10: & (f - g)(10) &= (10)^2 + (10) - 4 = 100 + 10 - 4 = 106 \\
\end{aligned}
\][/tex]
So, the results for [tex]\((f - g)(x)\)[/tex] from [tex]\(x = -10\)[/tex] to [tex]\(x = 10\)[/tex] are:
[tex]\[
[(-10, 86), (-9, 68), (-8, 52), (-7, 38), (-6, 26), (-5, 16), (-4, 8), (-3, 2), (-2, -2), (-1, -4), (0, -4), (1, -2), (2, 2), (3, 8), (4, 16), (5, 26), (6, 38), (7, 52), (8, 68), (9, 86), (10, 106)]
\][/tex]
This comprehensive evaluation list provides the values of [tex]\((f - g)(x)\)[/tex] for [tex]\(x\)[/tex] from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex].
