To find [tex]\((g - f)(x)\)[/tex], we need to subtract [tex]\(f(x)\)[/tex] from [tex]\(g(x)\)[/tex].
The functions given are:
[tex]\( f(x) = x^3 - 5x + 1 \)[/tex] defined on the interval [tex]\([-8, 5)\)[/tex]
[tex]\( g(x) = 4x^3 - 10x \)[/tex] defined on the interval [tex]\((2, 12]\)[/tex]
First, we write the expressions for [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex]:
[tex]\(f(x) = x^3 - 5x + 1\)[/tex]
[tex]\(g(x) = 4x^3 - 10x\)[/tex]
To find [tex]\((g - f)(x)\)[/tex], we subtract [tex]\(f(x)\)[/tex] from [tex]\(g(x)\)[/tex]:
[tex]\[
(g - f)(x) = g(x) - f(x)
\][/tex]
Substitute the given functions [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex]:
[tex]\[
(g - f)(x) = (4x^3 - 10x) - (x^3 - 5x + 1)
\][/tex]
Distribute the negative sign across the terms in [tex]\(f(x)\)[/tex]:
[tex]\[
(g - f)(x) = 4x^3 - 10x - x^3 + 5x - 1
\][/tex]
Combine like terms:
[tex]\[
(g - f)(x) = (4x^3 - x^3) + (-10x + 5x) - 1
\][/tex]
Simplify the expressions:
[tex]\[
(g - f)(x) = 3x^3 - 5x - 1
\][/tex]
Thus, the correct expression for [tex]\((g - f)(x)\)[/tex] is:
[tex]\[
(g - f)(x) = 3x^3 - 5x - 1
\][/tex]
The correct option among the given choices is:
[tex]\[
(g - f)(x) = 3x^3 - 5x - 1
\][/tex]
