If `f(x)=3x^(3)-2x^(2)+x-2` , and ` i =sqrt(-1)` then `f(i)` =

Options

A. `-2i-4`
B. 4i-4
C. 4i
D. `-2i`

Explanation

To find \( f(i) \) for the function \( f(x) = 3x^3 - 2x^2 + x - 2 \), where \( i = \sqrt{-1} \), we will substitute \( i \) into the function and simplify.

### Step-by-step Solution:

1. **Substitute \( i \) into the function**:
\[ f(i) = 3(i)^3 - 2(i)^2 + i - 2 \]

2. **Calculate \( i^2 \) and \( i^3 \)**:
- We know that \( i^2 = -1 \).
- To find \( i^3 \), we can use \( i^3 = i \cdot i^2 = i \cdot (-1) = -i \).

3. **Substitute \( i^2 \) and \( i^3 \) into the function**:
\[ f(i) = 3(-i) - 2(-1) + i - 2 \]

4. **Simplify the expression**:
\[ f(i) = -3i + 2 + i - 2 \]

5. **Combine like terms**:
- The constant terms \( +2 \) and \( -2 \) cancel each other out:
\[ f(i) = -3i + i = -2i \]

Thus, the final result is:
\[ f(i) = -2i \]