To find \((f-g)(x)\), we need to subtract the function \(g(x)\) from the function \(f(x)\).
Given:
[tex]\[f(x) = 4x^3 + 6x^2 - 3x - 4\][/tex]
[tex]\[g(x) = 4x - 3\][/tex]
Now, we calculate \((f-g)(x)\):
[tex]\[
(f-g)(x) = f(x) - g(x)
\][/tex]
[tex]\[
(f-g)(x) = (4x^3 + 6x^2 - 3x - 4) - (4x - 3)
\][/tex]
To perform the subtraction, distribute the negative sign through \(g(x)\):
[tex]\[
f(x) - g(x) = (4x^3 + 6x^2 - 3x - 4) - 4x + 3
\][/tex]
Now, combine like terms:
- Combine the \(x^3\) terms:
[tex]\[
4x^3
\][/tex]
- Combine the \(x^2\) terms:
[tex]\[
+6x^2
\][/tex]
- Combine the \(x\) terms:
[tex]\[
-3x - 4x = -7x
\][/tex]
- Combine the constant terms:
[tex]\[
-4 + 3 = -1
\][/tex]
Thus, \((f-g)(x)\) becomes:
[tex]\[
(f-g)(x) = 4x^3 + 6x^2 - 7x - 1
\][/tex]
Matching this result to the given options:
A. \((f-g)(x) = 4 x^3 + 6 x^2 + x - 7\)
B. \((f-g)(x) = 4 x^3 + 6 x^2 + x - 1\)
C. \((f-g)(x) = 4 x^3 + 6 x^2 - 7 x - 1\)
D. \((f-g)(x) = 4 x^3 + 6 x^2 - 7 x - 7\)
The correct answer is:
[tex]\[
\boxed{C. (f-g)(x) = 4 x^3 + 6 x^2 - 7 x - 1}
\][/tex]
