To find [tex]\( f(x) - g(x) \)[/tex] where [tex]\( f(x) = \sqrt{x} - x \)[/tex] and [tex]\( g(x) = 2x^3 - \sqrt{x} - x \)[/tex], we can proceed as follows:
1. Express [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] clearly:
- [tex]\( f(x) = \sqrt{x} - x \)[/tex]
- [tex]\( g(x) = 2x^3 - \sqrt{x} - x \)[/tex]
2. Form the expression [tex]\( f(x) - g(x) \)[/tex]:
[tex]\[
f(x) - g(x) = (\sqrt{x} - x) - (2x^3 - \sqrt{x} - x)
\][/tex]
3. Distribute the minus sign across the [tex]\( g(x) \)[/tex] terms:
[tex]\[
f(x) - g(x) = \sqrt{x} - x - 2x^3 + \sqrt{x} + x
\][/tex]
4. Combine like terms:
- Combine the [tex]\(\sqrt{x}\)[/tex] terms:
[tex]\[
\sqrt{x} + \sqrt{x} = 2\sqrt{x}
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-x + x = 0
\][/tex]
- The [tex]\(2x^3\)[/tex] term remains as it is:
[tex]\[
-2x^3
\][/tex]
5. Write the simplified expression:
[tex]\[
2\sqrt{x} - 2x^3
\][/tex]
So, the expression [tex]\( f(x) - g(x) \)[/tex] simplifies to:
[tex]\[
2\sqrt{x} - 2x^3
\][/tex]
From the provided choices:
A. [tex]\( 2x^3 - 2x + 2\sqrt{x} \)[/tex]
B. [tex]\( -2x^3 + 2\sqrt{x} \)[/tex]
C. [tex]\( -2x^3 - 2x \)[/tex]
D. [tex]\( -2x^3 - 2x - 2\sqrt{x} \)[/tex]
The correct choice is:
B. [tex]\( -2x^3 + 2\sqrt{x} \)[/tex]
Therefore, [tex]\( f(x) - g(x) = 2\sqrt{x} - 2x^3 \)[/tex].
\boxed{B}
