https://file.aiquickdraw.com/mathbot/file/1726115878948751uyudl.jpg

The function given in the image is:

\[
f(x) = \frac{x^2 - 1}{x^2 + 3x - 4}
\]

### Part (a) - Where is \( f(x) \) discontinuous?

To find where \( f(x) \) is discontinuous, we need to examine the denominator \( x^2 + 3x - 4 \). Discontinuities occur when the denominator equals zero because division by zero is undefined.

So, let's solve:

\[
x^2 + 3x - 4 = 0
\]

Factoring the quadratic:

\[
(x + 4)(x - 1) = 0
\]

This gives us the critical points \( x = -4 \) and \( x = 1 \), where \( f(x) \) is potentially discontinuous.

### Part (b) - Classify the type of discontinuity

To classify the discontinuity, we must check the behavior of the function around these points.

#### 1. At \( x = 1 \):

The numerator is \( x^2 - 1 = (x - 1)(x + 1) \), so at \( x = 1 \), the numerator also becomes zero. This creates a **removable discontinuity** at \( x = 1 \) because the common factor \( (x - 1) \) cancels out.

After canceling the common factor, the function simplifies to:

\[
f(x) = \frac{x + 1}{x + 4} \quad \text{for} \, x \neq 1
\]

Thus, the discontinuity at \( x = 1 \) is **removable**.

#### 2. At \( x = -4 \):

At \( x = -4 \), the denominator becomes zero, but the numerator does not. This causes a **non-removable (infinite) discontinuity**, also known as a **vertical asymptote**, at \( x = -4 \).

### Conclusion:
- \( f(x) \) is discontinuous at \( x = 1 \) (removable discontinuity).
- \( f(x) \) has an infinite discontinuity (vertical asymptote) at \( x = -4 \).

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Would you like more details on any of the steps? Here are some related questions:

1. How do we determine the limit of a function around a removable discontinuity?
2. What is the difference between removable and non-removable discontinuities?
3. How can you factor more complex polynomials to find discontinuities?
4. How does the behavior near a vertical asymptote affect the function graph?
5. What happens to the simplified function \( \frac{x+1}{x+4} \) as \( x \to -4 \)?

**Tip:** When checking for discontinuities, always verify whether the numerator and denominator share a common factor to identify removable discontinuities.

Given f(x) = (x^2 - 1) / (x^2 + 3x - 4), where is f(x) discontinuous? Classify the type of discontinuity.

Understand where the rational function f(x) = (x^2 - 1) / (x^2 + 3x - 4) is discontinuous and how to classify those discontinuities. Learn about removable and non-removable discontinuities by factoring and simplifying functions. Ideal for Grades 10-12.

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