We are tasked with evaluating the function:

\[
f(x) = \left[ (x - 2)^2 - x \right]^2
\]

at \( x = 3 \). Let's go step by step:

### Step 1: Substitute \( x = 3 \) into the expression inside the brackets.

\[
f(3) = \left[ (3 - 2)^2 - 3 \right]^2
\]

### Step 2: Simplify the expression inside the parentheses.

\[
(3 - 2)^2 = 1^2 = 1
\]

So, we have:

\[
f(3) = \left[ 1 - 3 \right]^2 = (-2)^2
\]

### Step 3: Square the result.

\[
(-2)^2 = 4
\]

Thus, \( f(3) = 4 \).

### Final Answer:
\[
f(3) = 4
\]

Would you like further details or explanations?

Here are 5 related questions:
1. How would you compute the derivative of \( f(x) \)?
2. What happens to \( f(x) \) as \( x \to 2 \)?
3. What is the second derivative of \( f(x) \)?
4. Can you sketch the graph of \( f(x) \)?
5. How would you find the critical points of \( f(x) \)?

**Tip**: Always simplify expressions inside parentheses or brackets first before applying further operations like squaring or taking powers.

Learn how to evaluate the function f(x) = [(x − 2)^2 − x]^2 at x = 3 with a step-by-step solution. This problem covers algebraic concepts including polynomial manipulation and exponents. Suitable for students in Grades 9-11.

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