The problem you're working with involves the rational function:

$f(x) = \frac{x}{x - 3}$

Let’s go through each part of the problem.

A. Find the following:

1. Domain: The domain of a function is all the possible values of $x$ for which the function is defined. For rational functions, the function is undefined when the denominator is zero. So, we find where $x - 3 = 0$, which gives $x = 3$. Therefore, the domain is: $\text{Domain: } (-\infty, 3) \cup (3, \infty)$

2. Range: To find the range of the function, we need to understand the behavior of $f(x)$. For this rational function, notice that as $x \to \infty$ or $x \to -\infty$, $f(x)$ approaches 1, but it never equals 1. Additionally, no value of $x$ makes $f(x) = 1$ exactly, since the horizontal asymptote is at $y = 1$. Therefore, the range is all real numbers except 1: $\text{Range: } (-\infty, 1) \cup (1, \infty)$

3. Vertical Asymptote: The vertical asymptote occurs where the denominator is zero, which we found to be $x = 3$. Therefore, the vertical asymptote is: $x = 3$

4. Horizontal Asymptote: For rational functions, the horizontal asymptote is determined by comparing the degrees of the numerator and denominator. Both the numerator and denominator have degree 1. The horizontal asymptote is the ratio of the leading coefficients of $x$, which is 1: $y = 1$

5. $x$-intercept: The $x$-intercept occurs where the function equals zero, i.e., where the numerator is zero. Set $x = 0$ in the numerator: $x = 0$ So, the $x$-intercept is: $(0, 0)$

6. $y$-intercept: The $y$-intercept occurs where $x = 0$. Substituting $x = 0$ into the function: $f(0) = \frac{0}{0 - 3} = 0$ So, the $y$-intercept is: $(0, 0)$

7. Table of values: Construct a table by selecting several values of $x$ around the vertical asymptote $x = 3$, for instance: $x = 1, 2, 4, 5$. We can calculate the corresponding $y$-values by substituting these values into the function.

B. Sketch the graph:

To sketch the graph:

• Include the vertical asymptote at $x = 3$.
• Include the horizontal asymptote at $y = 1$.
• Plot the intercepts at $(0, 0)$.
• Use the table of values to plot points and show the behavior of the graph near the asymptotes.

Let me know if you'd like me to calculate the table of values and generate a sketch for the graph!

Relative Questions:

1. How does the vertical asymptote affect the behavior of the function near $x = 3$?
2. Why is the horizontal asymptote located at $y = 1$?
3. How can you verify if a point is an intercept for the function?
4. What happens to the value of $f(x)$ as $x$ approaches very large or very small values?
5. Can the function $f(x)$ ever reach its horizontal asymptote?

Tip:

For rational functions, always check the degree of the numerator and denominator to easily find the horizontal asymptote.