Given the functions [tex]\( f(x) = 5x^2 - 3x + 1 \)[/tex] and [tex]\( g(x) = 2x^2 + x - 2 \)[/tex], perform the indicated operation.

Find [tex]\((f-g)(x)\)[/tex].

A. [tex]\( 3x^2 - 4x + 3 \)[/tex]

B. [tex]\( 3x^2 - 2x - 1 \)[/tex]

C. [tex]\( 3x^2 - 4x - 1 \)[/tex]

D. [tex]\( 3x^2 - 2x + 3 \)[/tex]

Note: State any domain restrictions if applicable.



Answer :

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]