https://file.aiquickdraw.com/mathbot/file/17255077404432kt0rre9.jpg

The function shown in the image is:

\[
F(x) = \frac{x^2 - 8x + 7}{x^2 - 5x - 14}
\]

To simplify and analyze this rational function, we can follow a few steps:

### Step 1: Factor both the numerator and the denominator.

**Numerator:**
\[
x^2 - 8x + 7
\]
This factors as:
\[
(x - 7)(x - 1)
\]

**Denominator:**
\[
x^2 - 5x - 14
\]
This factors as:
\[
(x - 7)(x + 2)
\]

### Step 2: Simplify the function.
The common factor \((x - 7)\) appears in both the numerator and denominator, so we can cancel it out, leaving:
\[
F(x) = \frac{x - 1}{x + 2}, \quad \text{for} \ x \neq 7
\]
The restriction \(x \neq 7\) comes from the fact that the factor \(x - 7\) in the denominator would make the function undefined at \(x = 7\).

### Step 3: Identify any restrictions.
- The function is undefined at \(x = 7\) because of the canceled factor.
- The function is also undefined at \(x = -2\) because the denominator \(x + 2 = 0\) at this point.

Thus, the domain of \(F(x)\) is all real numbers except \(x = 7\) and \(x = -2\).

Would you like more details on any part of this solution or further analysis?

Here are five related questions to expand on this topic:
1. How do you find vertical asymptotes for this rational function?
2. What happens to the function \(F(x)\) as \(x\) approaches infinity?
3. How can you determine if there are any holes in the graph of the function?
4. What are the intercepts of this simplified function?
5. How would you graph this rational function?

**Tip:** Always check for common factors in rational functions to simplify them, but remember to note any restrictions from canceled terms.

Simplify and analyze the rational function F(x) = (x^2 - 8x + 7) / (x^2 - 5x - 14) by factoring and identifying restrictions. Explore domain limitations and how to cancel common factors for a clearer function representation. Suitable for students in Grades 9-12.

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