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]
