To solve the problem of finding [tex]\((f - g)(x)\)[/tex] given the functions [tex]\(f(x) = 5x^2 - 3x + 1\)[/tex] and [tex]\(g(x) = 2x^2 + x - 2\)[/tex], let's go through the detailed step-by-step process:
1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 5x^2 - 3x + 1
\][/tex]
[tex]\[
g(x) = 2x^2 + x - 2
\][/tex]
2. Set up the operation [tex]\((f - g)(x)\)[/tex]:
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
3. Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f - g)(x) = (5x^2 - 3x + 1) - (2x^2 + x - 2)
\][/tex]
4. Distribute the subtraction across the terms inside the parentheses:
[tex]\[
(f - g)(x) = 5x^2 - 3x + 1 - 2x^2 - x + 2
\][/tex]
5. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(5x^2 - 2x^2 = 3x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-3x - x = -4x\)[/tex]
- Combine the constant terms: [tex]\(1 + 2 = 3\)[/tex]
Therefore,
[tex]\[
(f - g)(x) = 3x^2 - 4x + 3
\][/tex]
6. Write the final expression for [tex]\((f - g)(x)\)[/tex]:
[tex]\[
(f - g)(x) = 3x^2 - 4x + 3
\][/tex]
So, the correct expression for [tex]\((f - g)(x)\)[/tex] is:
[tex]\[
3x^2 - 4x + 3
\][/tex]
Domain Restriction:
Since both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are polynomials, they are defined for all real numbers. Therefore, the domain of [tex]\((f - g)(x)\)[/tex] is also all real numbers, [tex]\(\mathbb{R}\)[/tex].
Based on the computed expression, the correct choice from the given options is:
[tex]\[
3x^2 - 4x + 3
\][/tex]
