To determine where the function $f(x) = \frac{3x^2 - 6x + 9}{x^2 + x - 2}$ is undefined, we need to identify where the denominator equals zero, since division by zero is undefined.

Let's find the values of $x$ that make the denominator zero:

$x^2 + x - 2 = 0$

We can solve this quadratic equation using factoring. First, factor the quadratic:

$x^2 + x - 2 = (x - 1)(x + 2) = 0$

Now, set each factor equal to zero:

$x - 1 = 0 \quad \text{or} \quad x + 2 = 0$

Solving these:

$x = 1 \quad \text{or} \quad x = -2$

Therefore, the function is undefined at $x = 1$ and $x = -2$.

Summary:

The function $f(x) = \frac{3x^2 - 6x + 9}{x^2 + x - 2}$ is undefined at $x = 1$ and $x = -2$.

Would you like more details on how to solve this, or have any other questions?

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Here are 5 related questions to explore further:

1. How do you determine where a rational function has vertical asymptotes?
2. What happens to the function when the denominator has repeated roots?
3. Can you simplify this rational function?
4. How do you factor quadratics when solving for values of $x$?
5. What are the implications for the domain of a function when there are undefined points?

Tip: Always check the denominator for values that could make it zero when working with rational functions!