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If [tex]f(x) = x^3 - 2x^2[/tex], which expression is equivalent to [tex]f(i)[/tex]?

A. [tex]-2 + i[/tex]
B. [tex]-2 - i[/tex]
C. [tex]2 + i[/tex]
D. [tex]2 - i[/tex]

Sagot :

GinnyAnswer
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]
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