Answer:
To find \( f(f(x)) \), where \( f(x) = x^2 - 3x + 2 \), we first need to compute \( f(f(x)) \) step by step.
1. **Compute \( f(x) \):**
\[ f(x) = x^2 - 3x + 2 \]
2. **Substitute \( f(x) \) into itself:**
\[ f(f(x)) = f(x^2 - 3x + 2) \]
3. **Calculate \( f(x^2 - 3x + 2) \):**
Substitute \( x^2 - 3x + 2 \) into \( f(x) \):
\[ f(x^2 - 3x + 2) = (x^2 - 3x + 2)^2 - 3(x^2 - 3x + 2) + 2 \]
4. **Expand and simplify \( (x^2 - 3x + 2)^2 \):**
\[ (x^2 - 3x + 2)^2 = (x^2 - 3x + 2)(x^2 - 3x + 2) \]
\[ = x^4 - 6x^3 + 13x^2 - 12x + 4 \]
5. **Now substitute back into \( f(x^2 - 3x + 2) \):**
\[ f(x^2 - 3x + 2) = x^4 - 6x^3 + 13x^2 - 12x + 4 - 3(x^2 - 3x + 2) + 2 \]
\[ = x^4 - 6x^3 + 13x^2 - 12x + 4 - 3x^2 + 9x - 6 + 2 \]
\[ = x^4 - 6x^3 + 10x^2 - 3x \]
Therefore, \( f(f(x)) = x^4 - 6x^3 + 10x^2 - 3x \).
This is the function \( f(f(x)) \) defined by the original function \( f(x) = x^2 - 3x + 2 \).
