You are given the function $F(x) = \frac{x + 10}{9x - 1}$. Let's analyze this function step by step:

1. Domain:

The domain of a rational function is all real numbers except where the denominator is zero.

Set the denominator equal to zero to find the values that are not in the domain: $9x - 1 = 0$ Solve for $x$: $x = \frac{1}{9}$

Thus, the domain of the function is all real numbers except $x = \frac{1}{9}$. So: $\text{Domain: } x \in \mathbb{R}, x \neq \frac{1}{9}$

2. Vertical Asymptote:

A vertical asymptote occurs where the denominator is zero, as the function tends to infinity near those values. From the domain analysis, the vertical asymptote is at: $x = \frac{1}{9}$

3. Horizontal Asymptote:

To find the horizontal asymptote, examine the degrees of the polynomials in the numerator and denominator.

• The degree of the numerator ($x + 10$) is 1.
• The degree of the denominator ($9x - 1$) is also 1.

When the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 9. Thus, the horizontal asymptote is: $y = \frac{1}{9}$

4. Intercepts:

• x-intercept: Set $F(x) = 0$ (i.e., the numerator equals zero). $x + 10 = 0 \quad \Rightarrow \quad x = -10$ Thus, the x-intercept is at $(-10, 0)$.

• y-intercept: Set $x = 0$ and solve for $F(0)$: $F(0) = \frac{0 + 10}{9(0) - 1} = \frac{10}{-1} = -10$ Thus, the y-intercept is at $(0, -10)$.

5. Behavior near Asymptotes:

• As $x \to \frac{1}{9}$, the function approaches a vertical asymptote.
• As $x \to \infty$ or $x \to -\infty$, the function approaches the horizontal asymptote $y = \frac{1}{9}$.

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Do you need further clarification or want more details on any of these steps? Let me know!

Here are 5 related questions to deepen your understanding:

1. How do you find slant asymptotes for rational functions?
2. What happens to the graph of $F(x)$ as $x$ approaches the vertical asymptote?
3. Can a function have more than one vertical asymptote?
4. How does the behavior of rational functions differ depending on the degrees of the numerator and denominator?
5. What would happen if the numerator had a higher degree than the denominator?

Tip: To graph rational functions effectively, always check for domain restrictions, intercepts, and asymptotic behavior!