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Let's break down the problem step by step. The function given is:

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

### Step 1: Factor the denominator and numerator
First, factor both the numerator and the denominator to analyze the discontinuities.

**Denominator:**
\[
x^2 + x - 2 = (x - 1)(x + 2)
\]

**Numerator:**
\[
3x^2 + 4x - 4 = (3x - 2)(x + 2)
\]

### Step 2: Analyze the discontinuities
Discontinuities occur when the denominator equals zero, so let's solve for \(x\) when the denominator is zero.

\[
(x - 1)(x + 2) = 0
\]
This gives us:
\[
x = 1 \quad \text{or} \quad x = -2
\]

At these points, we need to determine whether the discontinuities are removable or vertical.

### i. Removable discontinuity at \(x = -2\)?

At \(x = -2\), both the numerator \((x + 2)\) and denominator \((x + 2)\) have the factor \(x + 2\). This suggests a **removable discontinuity** at \(x = -2\), because the factor cancels out.

**Answer**: True

### ii. Vertical discontinuity at \(x = 1\)?

At \(x = 1\), the denominator has a factor \((x - 1)\), but the numerator does not. This means that there is a **vertical asymptote** at \(x = 1\), which indicates a vertical discontinuity.

**Answer**: True

### iii. Removable discontinuity at \(x = \frac{2}{3}\)?

At \(x = \frac{2}{3}\), the value does not make the denominator zero, so there is no discontinuity at this point. Therefore, there is no removable discontinuity at \(x = \frac{2}{3}\).

**Answer**: False

### Final Answers:
i. True  
ii. True  
iii. False

Would you like more details or have any questions?

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Here are 5 follow-up questions related to this problem:

1. How does factoring help identify discontinuities in rational functions?
2. What is the difference between removable and vertical discontinuities?
3. How would the graph of the function behave around \(x = 1\) and \(x = -2\)?
4. Can a function have both removable and vertical discontinuities at the same point?
5. How would the function be redefined to remove the discontinuity at \(x = -2\)?

**Tip**: Always check for common factors in both the numerator and denominator to determine whether discontinuities are removable!

Consider the function f(x) = (3x^2 + 4x - 4) / (x^2 + x - 2). Which of the following statements are true?

i. There is a removable discontinuity at x = -2

ii. There is a vertical discontinuity at x = 1

iii. There is a removable discontinuity at x = 2/3

Learn how to identify and classify discontinuities in the rational function f(x) = (3x^2 + 4x - 4) / (x^2 + x - 2). This problem explores removable and vertical discontinuities and provides a detailed step-by-step breakdown. Suitable for students in Grades 11-12.

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