The given function from the image is:

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

This is a rational function, meaning it is a ratio of two polynomials. Let's simplify it if possible and discuss its properties, such as domain restrictions.

Step 1: Factorize the numerator and denominator

• Numerator: $x^2 - 3x + 2$

  • This factors as $(x - 1)(x - 2)$.

• Denominator: $x^2 - 4$

  • This is a difference of squares, which factors as $(x - 2)(x + 2)$.

So, the function becomes: $f(x) = \frac{(x - 1)(x - 2)}{(x - 2)(x + 2)}$

Step 2: Cancel common factors

The term $(x - 2)$ appears in both the numerator and the denominator, so we can cancel it out, provided $x \neq 2$ (because division by zero is undefined).

This simplifies to: $f(x) = \frac{x - 1}{x + 2}, \quad \text{for} \ x \neq 2$

Step 3: Domain restrictions

We have two values where the denominator of the original function becomes zero:

• $x = 2$
• $x = -2$

Thus, the domain of the function is all real numbers except $x = 2$ and $x = -2$.

Final simplified form:

$f(x) = \frac{x - 1}{x + 2}, \quad \text{with} \ x \neq 2, \ x \neq -2$

Would you like to explore this function further, such as finding vertical and horizontal asymptotes or graphing the function? Let me know if you have any questions!

Here are 5 relative questions based on the function simplification:

1. What are the vertical asymptotes of the simplified function?
2. What is the horizontal asymptote of this rational function?
3. How would you find the x-intercept and y-intercept of this function?
4. Can you explain why $x = 2$ is a removable discontinuity?
5. How would you describe the behavior of this function as $x \to \infty$?

Tip: Always remember to check for domain restrictions before simplifying any rational expression to avoid undefined values.