To solve for [tex]\((f-g)(x)\)[/tex], we need to subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex].
Given the functions:
[tex]\[ f(x) = 3x^3 + 4x^2 - 6x - 7 \][/tex]
[tex]\[ g(x) = 2x - 4 \][/tex]
We can write the subtraction of [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex] as follows:
[tex]\[ (f-g)(x) = f(x) - g(x) \][/tex]
Now, let's perform the subtraction step-by-step:
1. Write down [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] in their expanded forms:
[tex]\[ f(x) = 3x^3 + 4x^2 - 6x - 7 \][/tex]
[tex]\[ g(x) = 2x - 4 \][/tex]
2. Subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]:
[tex]\[ (f-g)(x) = (3x^3 + 4x^2 - 6x - 7) - (2x - 4) \][/tex]
3. Distribute the negative sign across [tex]\(g(x)\)[/tex]:
[tex]\[ (f-g)(x) = 3x^3 + 4x^2 - 6x - 7 - 2x + 4 \][/tex]
4. Combine like terms:
[tex]\[ (f-g)(x) = 3x^3 + 4x^2 - 6x - 2x - 7 + 4 \][/tex]
[tex]\[ (f-g)(x) = 3x^3 + 4x^2 - 8x - 3 \][/tex]
So, the function [tex]\((f-g)(x)\)[/tex] is:
[tex]\[ (f-g)(x) = 3x^3 + 4x^2 - 8x - 3 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D. \ (f-g)(x) = 3x^3 + 4x^2 - 8x - 3} \][/tex]
