List all the values of x for which the function f x =
x
2−2x+1
2x
2−7x+5

is not

continuous.

We are given the function \( f(x) = \frac{x^2 - 2x + 1}{2x^2 - 7x + 5} \), and we need to find the values of \(x\) for which the function is not continuous.

To determine where the function is not continuous, we need to find when the denominator is equal to zero, because division by zero is undefined and causes discontinuity.

### Step 1: Set the denominator equal to zero
We need to solve the equation:
\[
2x^2 - 7x + 5 = 0
\]

### Step 2: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 2x^2 - 7x + 5 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -7 \)
- \( c = 5 \)

Substituting these values into the quadratic formula:
\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(5)}}{2(2)}
\]
\[
x = \frac{7 \pm \sqrt{49 - 40}}{4}
\]
\[
x = \frac{7 \pm \sqrt{9}}{4}
\]
\[
x = \frac{7 \pm 3}{4}
\]

### Step 3: Find the two solutions
\[
x = \frac{7 + 3}{4} = \frac{10}{4} = 2.5
\]
\[
x = \frac{7 - 3}{4} = \frac{4}{4} = 1
\]

### Step 4: Conclusion
The function \( f(x) \) is not continuous at \( x = 1 \) and \( x = 2.5 \), because the denominator equals zero at these points, making the function undefined.

So, the values of \( x \) for which the function is not continuous are:
\[
x = 1 \quad \text{and} \quad x = 2.5
\]

Would you like a detailed explanation of how the quadratic formula was applied or have any other questions?

Here are 5 related questions:
1. How do you determine if a rational function is continuous?
2. What happens to a function at points where the denominator equals zero?
3. Can you explain why a function might have a removable discontinuity?
4. What are the different types of discontinuities in functions?
5. How would this function behave near \( x = 1 \) and \( x = 2.5 \)?

**Tip**: When solving quadratic equations, always check the discriminant \( b^2 - 4ac \) first to know if real solutions exist.

List all the values of x for which the function f(x) = (x^2 - 2x + 1) / (2x^2 - 7x + 5) is not continuous.

Discover how to determine the values of x where the rational function f(x) = (x^2 - 2x + 1) / (2x^2 - 7x + 5) is not continuous. Learn key concepts such as solving quadratic equations and identifying discontinuities. Ideal for students in Grades 10-12.

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