`f(x)=x^(3)-(k-2)x^(2)+2x`, for all x and if it is an odd function, find k.

Answer

To determine the value of \( k \) for the function \( f(x) = x^3 - (k-2)x^2 + 2x \) to be an odd function, we will follow these steps:

### Step 1: Understand the property of odd functions
An odd function satisfies the property:
\[ f(-x) = -f(x) \quad \text{for all } x \]

### Step 2: Calculate \( f(-x) \)
We need to find \( f(-x) \):
\[ f(-x) = (-x)^3 - (k-2)(-x)^2 + 2(-x) \]
Calculating each term:
\[ f(-x) = -x^3 - (k-2)x^2 - 2x \]
Thus, we have:
\[ f(-x) = -x^3 - (k-2)x^2 - 2x \]

### Step 3: Calculate \( -f(x) \)
Now, we compute \( -f(x) \):
\[ -f(x) = -\left(x^3 - (k-2)x^2 + 2x\right) \]
This simplifies to:
\[ -f(x) = -x^3 + (k-2)x^2 - 2x \]

### Step 4: Set \( f(-x) \) equal to \( -f(x) \)
For \( f(x) \) to be an odd function, we need:
\[ f(-x) = -f(x) \]
Substituting the expressions we derived:
\[ -x^3 - (k-2)x^2 - 2x = -x^3 + (k-2)x^2 - 2x \]

### Step 5: Simplify the equation
Now, we can simplify the equation:
\[ -x^3 - (k-2)x^2 - 2x = -x^3 + (k-2)x^2 - 2x \]
Cancelling \( -x^3 \) and \( -2x \) from both sides gives:
\[ -(k-2)x^2 = (k-2)x^2 \]

### Step 6: Solve for \( k \)
This implies:
\[ -(k-2) = (k-2) \]
Adding \( (k-2) \) to both sides results in:
\[ 0 = 2(k-2) \]
This simplifies to:
\[ k - 2 = 0 \]
Thus, we find:
\[ k = 2 \]

### Conclusion
The value of \( k \) that makes the function \( f(x) \) an odd function is:
\[ \boxed{2} \]