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Select all the correct answers.

If [tex]f(x) = x^2 - 3x - 4[/tex] and [tex]g(x) = x^2 + x[/tex], what is [tex](f+g)(x)[/tex]?

A. [tex]2x^2 - 2x - 4[/tex]

B. [tex]2(x^2 - x - 2)[/tex]

C. [tex]x^2 - x - 4[/tex]

D. [tex]x^2 - 1[/tex]



Answer :

GinnyAnswer
To find [tex]\((f + g)(x)\)[/tex] given [tex]\(f(x) = x^2 - 3x - 4\)[/tex] and [tex]\(g(x) = x^2 + x\)[/tex]:

1. Start by finding [tex]\(f(x) + g(x)\)[/tex]:

[tex]\[ f(x) = x^2 - 3x - 4 \][/tex]

[tex]\[ g(x) = x^2 + x \][/tex]

2. Add the two functions:

[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]

3. Substitute [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into the equation:

[tex]\[ (f + g)(x) = (x^2 - 3x - 4) + (x^2 + x) \][/tex]

4. Combine like terms:

[tex]\[ (f + g)(x) = x^2 + x^2 - 3x + x - 4 \][/tex]

Simplifying the expression:

[tex]\[ (f + g)(x) = 2x^2 - 2x - 4 \][/tex]

5. Therefore, [tex]\((f + g)(x) = 2x^2 - 2x - 4\)[/tex].

Next, let's consider the answer choices:

- [tex]\(2x^2 - 2x - 4\)[/tex]
- [tex]\(2(x^2 - x - 2)\)[/tex]
- [tex]\(x^2 - x - 4\)[/tex]
- [tex]\(x^2 - 1\)[/tex]

From our simplified expression, [tex]\((f + g)(x) = 2x^2 - 2x - 4\)[/tex]:

- The first choice, [tex]\(2x^2 - 2x - 4\)[/tex], is correct as it matches our simplified expression.
- The second choice, [tex]\(2(x^2 - x - 2)\)[/tex], is also correct because [tex]\(2(x^2 - x - 2)\)[/tex] simplifies to [tex]\(2x^2 - 2x - 4\)[/tex].

Thus, the correct answers are:
- [tex]\(2x^2 - 2x - 4\)[/tex]
- [tex]\(2(x^2 - x - 2)\)[/tex]

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