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**Problem 7:**
\[ F(x) = \frac{x+1}{x^2 - 3x - 4} \]
**Find \( F(0) \), \( F(-2) \), \( F(-4) \). Also, find the values of \( x \) for which \( F(x) \) is not defined.**

### Solution:
First, let's express the denominator of \( F(x) \) in a factored form to identify the values of \( x \) where \( F(x) \) is undefined.

\[ x^2 - 3x - 4 = (x - 4)(x + 1) \]

Thus, the function \( F(x) \) becomes:
\[ F(x) = \frac{x+1}{(x-4)(x+1)} \]

Here, \( F(x) \) is undefined when the denominator is zero, i.e., when \( x - 4 = 0 \) or \( x + 1 = 0 \). Therefore, \( F(x) \) is undefined at:
\[ x = 4 \text{ and } x = -1 \]

Now, we evaluate \( F(x) \) at the given points.

1. **Find \( F(0) \):**

   \[
   F(0) = \frac{0 + 1}{(0 - 4)(0 + 1)} = \frac{1}{(-4)(1)} = \frac{1}{-4} = -\frac{1}{4}
   \]

2. **Find \( F(-2) \):**

   \[
   F(-2) = \frac{-2 + 1}{(-2 - 4)(-2 + 1)} = \frac{-1}{(-6)(-1)} = \frac{-1}{6}
   \]

3. **Find \( F(-4) \):**

   \[
   F(-4) = \frac{-4 + 1}{(-4 - 4)(-4 + 1)} = \frac{-3}{(-8)(-3)} = \frac{-3}{24} = -\frac{1}{8}
   \]

### Summary:
- \( F(0) = -\frac{1}{4} \)
- \( F(-2) = -\frac{1}{6} \)
- \( F(-4) = -\frac{1}{8} \)
- \( F(x) \) is not defined at \( x = 4 \) and \( x = -1 \).

Would you like more detailed explanations or assistance with another problem from the image?

### Additional Related Questions:
1. How can you find the points where a function is undefined?
2. What is the importance of factoring in solving rational functions?
3. How do you interpret the results of \( F(x) \) when evaluating at different points?
4. What is the domain of the function \( F(x) \) given in the problem?
5. Can you graph the function \( F(x) = \frac{x+1}{x^2 - 3x - 4} \) and identify its asymptotes?

**Tip:** Always check for undefined points in a rational function by setting the denominator equal to zero and solving for \( x \).

If F(x) = (x + 1) / (x^2 - 3x - 4), find F(0), F(-2), F(-4). Also, find the values of x for which F(x) is not defined.

Learn how to solve the rational function F(x) = (x + 1) / (x^2 - 3x - 4). This detailed solution explains how to factor the quadratic expression in the denominator, evaluate the function at specific points, and find where the function is undefined. Suitable for students in Grades 10-12.

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