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We are tasked with studying the continuity of the function \( f(x) = \frac{2x - 4}{x^2 + x - 6} \).

To analyze the continuity of the function, we first check the points where the function might be undefined, which occur when the denominator is zero.

### Step 1: Factor the Denominator

The denominator is \( x^2 + x - 6 \). We need to factor this quadratic expression:

\[
x^2 + x - 6 = (x - 2)(x + 3)
\]

### Step 2: Identify Points of Discontinuity

The function \( f(x) \) will be discontinuous where the denominator is zero, so we solve:

\[
(x - 2)(x + 3) = 0
\]

This gives two solutions:
\[
x = 2 \quad \text{and} \quad x = -3
\]

These are the points where the function might have discontinuities.

### Step 3: Type of Discontinuity

- At \( x = 2 \), the numerator also becomes zero: \( 2x - 4 = 2(2) - 4 = 0 \), meaning there is a removable discontinuity at \( x = 2 \). This is because the factor \( (x - 2) \) cancels out from the numerator and denominator.

- At \( x = -3 \), the numerator does not become zero, meaning this is a **non-removable discontinuity** (specifically an **infinite discontinuity**) because the function tends to infinity as \( x \) approaches \( -3 \).

### Conclusion

The function \( f(x) \) is continuous everywhere except at:
- \( x = 2 \) (removable discontinuity),
- \( x = -3 \) (infinite discontinuity).

Would you like more details on how to confirm the type of discontinuity? Let me know if you have any questions! Here are five related questions to deepen your understanding:

1. How can you confirm the removable discontinuity algebraically?
2. What is the behavior of the function near \( x = -3 \)?
3. Can a function with a removable discontinuity be redefined to become continuous?
4. How do you calculate the limit of the function as \( x \to 2 \)?
5. What graphical features would help identify the types of discontinuities?

**Tip:** Always factor the denominator of rational functions to locate potential points of discontinuity.

Estudia la continuïtat de la funció f(x) = (2x-4) / (x^2 + x - 6). Si en algun punt la funció no és contínua, indica el tipus de discontinuïtat.

Explore the continuity of the function f(x) = (2x-4) / (x^2 + x - 6). Learn how to factor the denominator and determine points of discontinuity, including removable and infinite discontinuities. Suitable for Grades 10-12.

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