You are given the function \( f(x) = x^2 - 2x + 2 \), and you're asked to evaluate the function at \( f(2x - 3) \).

This means you need to substitute \( x \) in the function \( f(x) \) with \( 2x - 3 \). So, we replace every occurrence of \( x \) in the formula with \( 2x - 3 \).

Let's compute it:

\[
f(2x - 3) = (2x - 3)^2 - 2(2x - 3) + 2
\]

Now, expand and simplify the expression step by step.

### Step 1: Expand \( (2x - 3)^2 \)
\[
(2x - 3)^2 = (2x - 3)(2x - 3) = 4x^2 - 12x + 9
\]

### Step 2: Expand \( -2(2x - 3) \)
\[
-2(2x - 3) = -4x + 6
\]

### Step 3: Combine everything
\[
f(2x - 3) = 4x^2 - 12x + 9 - 4x + 6 + 2
\]

### Step 4: Simplify the expression
\[
f(2x - 3) = 4x^2 - 16x + 17
\]

Thus, the function evaluated at \( f(2x - 3) \) is:
\[
f(2x - 3) = 4x^2 - 16x + 17
\]

Would you like further details on this? Here are some related questions:

1. How do you evaluate functions at other variable substitutions?
2. Can we find the roots of the new function \( f(2x - 3) \)?
3. What is the vertex of the quadratic function \( f(2x - 3) \)?
4. How do transformations affect the graph of a function?
5. How do you compute the derivative of this function?

**Tip:** When substituting expressions into a function, always carefully replace every instance of the original variable with the new expression.

Learn how to evaluate the function f(x) = x² - 2x + 2 at f(2x - 3) step-by-step. Understand the process of substitution and simplification in quadratic functions. Suitable for high school students.

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