The function given in the image is:

$f(x) = \frac{x^2 + 2x - 8}{x^2 - 7x + 10}$

Let's analyze the function step by step:

1. Factorization:

We will attempt to factor both the numerator and the denominator to see if any simplifications can be made.

• Numerator: $x^2 + 2x - 8$

  • This quadratic can be factored as: $x^2 + 2x - 8 = (x + 4)(x - 2)$

• Denominator: $x^2 - 7x + 10$

  • This quadratic can be factored as: $x^2 - 7x + 10 = (x - 5)(x - 2)$

So the function becomes:

$f(x) = \frac{(x + 4)(x - 2)}{(x - 5)(x - 2)}$

2. Simplification:

Notice that $(x - 2)$ appears in both the numerator and denominator. Therefore, as long as $x \neq 2$, we can simplify the expression by canceling $(x - 2)$:

$f(x) = \frac{x + 4}{x - 5}, \quad \text{for} \quad x \neq 2$

3. Domain:

• The original function has restrictions on its domain due to the denominator. Specifically, the denominator $x^2 - 7x + 10$ cannot be zero, because division by zero is undefined.
• Setting the denominator equal to zero and solving: $x^2 - 7x + 10 = 0 \implies (x - 5)(x - 2) = 0$ Thus, $x = 5$ and $x = 2$ are the values where the denominator becomes zero. Therefore, $f(x)$ is undefined at $x = 5$ and $x = 2$.

So, the domain of $f(x)$ is: $x \in (-\infty, 2) \cup (2, 5) \cup (5, \infty)$

4. Conclusion:

The simplified function is:

$f(x) = \frac{x + 4}{x - 5}, \quad x \neq 2, x \neq 5$

Would you like further details or have any other questions?

Here are five related questions to deepen your understanding:

1. How would you find the vertical asymptotes of $f(x)$?
2. Can you determine the horizontal asymptote of the simplified function?
3. What would the graph of the function look like, considering the domain and asymptotes?
4. How would you solve $f(x) = 0$ for this function?
5. What are the critical points of $f(x)$ if you were to take its derivative?

Tip: Always check for values that make the denominator zero when simplifying rational functions, as these affect the domain and possible discontinuities.