Find f(x) if `f(x+1)=x^(2)-3x+2`

Answer

To find \( f(x) \) given that \( f(x+1) = x^2 - 3x + 2 \), we will follow these steps:

### Step 1: Substitute \( x \) with \( z - 1 \)
Let \( z = x + 1 \). Therefore, \( x = z - 1 \). We can rewrite the equation:
\[ f(z) = (z - 1)^2 - 3(z - 1) + 2 \]

### Step 2: Expand the equation
Now, we will expand the right-hand side:
\[ (z - 1)^2 = z^2 - 2z + 1 \]
\[ -3(z - 1) = -3z + 3 \]
Combining these, we have:
\[ f(z) = z^2 - 2z + 1 - 3z + 3 + 2 \]

### Step 3: Simplify the expression
Now, combine like terms:
\[ f(z) = z^2 - 5z + 6 \]

### Step 4: Replace \( z \) back with \( x \)
Since we initially let \( z = x + 1 \), we replace \( z \) back with \( x \):
\[ f(x) = x^2 - 5x + 6 \]

### Final Answer
Thus, the function \( f(x) \) is:
\[ \boxed{x^2 - 5x + 6} \]