Determine all values of *x* at which the function is discontinuous. (Enter your answers as a comma-separated list.)

*f*(*x*) = 

*x*2 − 4*x* + 4

*x*2 − 2*x*

*x* =  
 
 

To determine where the function \( f(x) \) is discontinuous, let's first rewrite the function clearly:

\[
f(x) = \frac{x^2 - 4x + 4}{x^2 - 2x}
\]

### Step 1: Factor the numerator and the denominator.

#### Numerator: \( x^2 - 4x + 4 \)
This is a perfect square trinomial, so we can factor it as:
\[
x^2 - 4x + 4 = (x - 2)^2
\]

#### Denominator: \( x^2 - 2x \)
We can factor this as:
\[
x^2 - 2x = x(x - 2)
\]

So the function can now be written as:
\[
f(x) = \frac{(x - 2)^2}{x(x - 2)}
\]

### Step 2: Simplify the function.
We can cancel out the common factor \( (x - 2) \) from both the numerator and denominator, but we must keep in mind that \( x - 2 = 0 \) when \( x = 2 \). This creates a potential issue at \( x = 2 \).

After cancellation, the simplified function is:
\[
f(x) = \frac{x - 2}{x}, \quad \text{for} \ x \neq 2
\]

### Step 3: Identify points of discontinuity.

#### 1. **At \( x = 0 \):**  
The denominator \( x \) becomes zero, so the function is undefined at \( x = 0 \).

#### 2. **At \( x = 2 \):**  
Although we canceled the factor \( x - 2 \), the original function had a zero in the denominator when \( x = 2 \), so the function is also undefined at \( x = 2 \).

### Conclusion:  
The function is discontinuous at \( x = 0 \) and \( x = 2 \).

Thus, the values of \( x \) at which the function is discontinuous are:

\[
\boxed{0, 2}
\]

Would you like more details or further clarifications?

Here are 5 related questions to deepen your understanding:
1. How do you determine if a function is continuous at a point?
2. What is the significance of factoring when simplifying rational functions?
3. How do you find the domain of a rational function?
4. What is the difference between removable and non-removable discontinuities?
5. How would you graph this function and show its discontinuities?

**Tip:** Always check for common factors that can cancel out, but remember to account for the points where the denominator equals zero.

Determine all values of *x* at which the function is discontinuous. (Enter your answers as a comma-separated list.) *f(x)* = *x*^2 − 4*x* + 4 / *x*^2 − 2*x*

This solution helps you determine the points of discontinuity for the rational function f(x) = (x^2 - 4x + 4)/(x^2 - 2x). Learn how factoring and identifying zeros in the denominator lead to discontinuities at x = 0 and x = 2. Suitable for high school students in Grades 10-12.

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