To solve the problem of finding [tex]\( f(x) - g(x) \)[/tex] given [tex]\( f(x) = x^{\frac{1}{2}} - x \)[/tex] and [tex]\( g(x) = 2x^3 - x^{\frac{1}{2}} - x \)[/tex], we need to follow these steps:
1. Write down the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = x^{\frac{1}{2}} - x
\][/tex]
[tex]\[
g(x) = 2x^3 - x^{\frac{1}{2}} - x
\][/tex]
2. Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) - g(x) = \left( x^{\frac{1}{2}} - x \right) - \left( 2x^3 - x^{\frac{1}{2}} - x \right)
\][/tex]
3. Distribute the negative sign across the [tex]\( g(x) \)[/tex] terms:
[tex]\[
f(x) - g(x) = x^{\frac{1}{2}} - x - 2x^3 + x^{\frac{1}{2}} + x
\][/tex]
4. Combine like terms:
- Combine the [tex]\( x^{\frac{1}{2}} \)[/tex] terms:
[tex]\[
x^{\frac{1}{2}} + x^{\frac{1}{2}} = 2x^{\frac{1}{2}}
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
-x + x = 0
\][/tex]
- Keep the term involving [tex]\( x^3 \)[/tex] as it is:
[tex]\[
-2x^3
\][/tex]
5. Write the simplified expression:
[tex]\[
f(x) - g(x) = 2x^{\frac{1}{2}} - 2x^3
\][/tex]
Therefore, the correct answer is:
[tex]\[ D. -2 x^3 + 2 x^{\frac{1}{2}} \][/tex]
