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If [tex][tex]$f(x) = x^3 - 2x^2 + 3x - 5$[/tex][/tex] and [tex][tex]$g(x) = x^2 + x - 1$[/tex][/tex], perform the given function operation.

Find [tex][tex]$f(x) + g(x)$[/tex][/tex].



Answer :

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]