The given function is $f(x) = \frac{1}{x - 1}$. This is a rational function, and we need to identify and draw the horizontal and vertical asymptotes based on this function.

Horizontal Asymptote:

For rational functions of the form $\frac{P(x)}{Q(x)}$, the horizontal asymptote depends on the degrees of the polynomials $P(x)$ and $Q(x)$:

• If the degree of $P(x)$ is less than the degree of $Q(x)$, the horizontal asymptote is $y = 0$.
• If the degrees are equal, the asymptote is $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$.
• If the degree of $P(x)$ is greater than $Q(x)$, there is no horizontal asymptote.

In this case, $f(x) = \frac{1}{x - 1}$, so the degree of the numerator is 0 (since it's a constant), and the degree of the denominator is 1. Thus, the horizontal asymptote is: $y = 0$ So, draw a horizontal line at $y = 0$ on the graph.

Vertical Asymptote:

The vertical asymptote occurs where the denominator equals zero, which would make the function undefined. Setting the denominator equal to zero: $x - 1 = 0$ $x = 1$ Thus, there is a vertical asymptote at $x = 1$.

Steps for Drawing:

1. Horizontal Asymptote: Draw a horizontal line at $y = 0$ (along the x-axis).
2. Vertical Asymptote: Draw a vertical line at $x = 1$.

This will complete the graph with the correct asymptotes.

Would you like further details or help with graphing?

Follow-up questions:

1. How do you identify vertical asymptotes for any rational function?
2. What happens to the graph of $f(x)$ as $x$ approaches 1?
3. Can a rational function have more than one horizontal asymptote?
4. How does the behavior near vertical asymptotes affect the overall shape of the graph?
5. What is the relationship between the degree of the numerator and denominator in determining horizontal asymptotes?

Tip: When graphing, always consider the behavior of the function as $x \to \infty$ and as $x \to \pm \infty$ to help identify asymptotes.