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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 $x = -1$.

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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 $x$.