To solve [tex]\( f(i) \)[/tex] where [tex]\( f(x) = x^3 - 2x^2 \)[/tex]:
1. Substitute [tex]\( i \)[/tex] (the imaginary unit, where [tex]\( i = \sqrt{-1} \)[/tex]) into the function:
[tex]\[
f(i) = i^3 - 2i^2
\][/tex]
2. Calculate [tex]\( i^3 \)[/tex]:
- Remember that [tex]\( i^2 = -1 \)[/tex].
- Thus, [tex]\( i^3 = i^2 \cdot i = (-1) \cdot i = -i \)[/tex].
3. Calculate [tex]\( 2i^2 \)[/tex]:
- Knowing that [tex]\( i^2 = -1 \)[/tex], then [tex]\( 2i^2 = 2 \cdot (-1) = -2 \)[/tex].
4. Combine the results:
[tex]\[
f(i) = -i - (-2) = -i + 2 = 2 - i
\][/tex]
So, the expression equivalent to [tex]\( f(i) \)[/tex] is [tex]\( 2 - i \)[/tex].
Therefore, the correct option is:
[tex]\[
\boxed{2-i}
\][/tex]
