$
\begin{array}{l}
f(x) = 4x^3 + 6x^2 - 3x - 4 \\
g(x) = 4x - 3
\end{array}
$

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

A. [tex]$(f-g)(x) = 4x^3 + 6x^2 + x - 7$[/tex]
B. [tex]$(f-g)(x) = 4x^3 + 6x^2 + x - 1$[/tex]
C. [tex]$(f-g)(x) = 4x^3 + 6x^2 - 7x - 1$[/tex]
D. [tex]$(f-g)(x) = 4x^3 + 6x^2 - 7x - 7$[/tex]



Answer :

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]