To find [tex]\(f(x) + g(x)\)[/tex] given the functions [tex]\(f(x) = x^3 - 2x^2 + 3x - 5\)[/tex] and [tex]\(g(x) = x^2 + x - 1\)[/tex], we need to perform the addition of these two functions.
Here are the steps:
1. Write down the functions:
[tex]\[
f(x) = x^3 - 2x^2 + 3x - 5
\][/tex]
[tex]\[
g(x) = x^2 + x - 1
\][/tex]
2. Perform the addition [tex]\(f(x) + g(x)\)[/tex] by combining like terms:
[tex]\[
f(x) + g(x) = (x^3 - 2x^2 + 3x - 5) + (x^2 + x - 1)
\][/tex]
3. Combine the corresponding terms for each power of [tex]\(x\)[/tex]:
[tex]\[
= x^3 + (-2x^2 + x^2) + (3x + x) + (-5 - 1)
\][/tex]
4. Simplify each group of like terms:
[tex]\[
x^3 + (-2x^2 + x^2) + (3x + x) + (-5 - 1)
\][/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-2x^2 + x^2 = -x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(3x + x = 4x\)[/tex]
- Combine constant terms: [tex]\(-5 - 1 = -6\)[/tex]
5. Write down the simplified result:
[tex]\[
f(x) + g(x) = x^3 - x^2 + 4x - 6
\][/tex]
6. Therefore, the final result of the function operation [tex]\(f(x) + g(x)\)[/tex] is:
[tex]\[
f(x) + g(x) = x^3 - x^2 + 4x - 6
\][/tex]
