https://file.aiquickdraw.com/mathbot/file/1726797216172ga2z0hzw.jpg

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?

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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.

Simplify the function f(x) = (x^2 + 2x - 8) / (x^2 - 7x + 10) and determine its domain.

Learn how to simplify the rational function f(x) = (x^2 + 2x - 8) / (x^2 - 7x + 10) through quadratic factorization and identify its domain. This problem is ideal for students in Grades 9-11, covering algebraic techniques and key concepts in rational functions.

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