To find [tex]\( f(x) - g(x) \)[/tex] given the functions [tex]\( f(x) = \sqrt{x} - x \)[/tex] and [tex]\( g(x) = 2x^3 - \sqrt{x} - x \)[/tex], follow these steps:
1. First, write down the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = \sqrt{x} - x
\][/tex]
[tex]\[
g(x) = 2x^3 - \sqrt{x} - x
\][/tex]
2. Now, compute [tex]\( f(x) - g(x) \)[/tex]:
[tex]\[
f(x) - g(x) = (\sqrt{x} - x) - (2x^3 - \sqrt{x} - x)
\][/tex]
3. Distribute the negative sign in front of [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) - g(x) = \sqrt{x} - x - 2x^3 + \sqrt{x} + x
\][/tex]
4. Combine like terms:
[tex]\[
f(x) - g(x) = (\sqrt{x} + \sqrt{x}) + (-x + x) - 2x^3
\][/tex]
[tex]\[
f(x) - g(x) = 2\sqrt{x} - 2x^3
\][/tex]
Thus, the simplified form of [tex]\( f(x) - g(x) \)[/tex] is:
[tex]\[
f(x) - g(x) = 2\sqrt{x} - 2x^3
\][/tex]
So the correct answer is:
A. [tex]\( -2x^3 + 2\sqrt{x} \)[/tex]
