The rational function given is $f(x) = \frac{1}{x - 1}$.

To determine the asymptotes:

1. Vertical Asymptote (V.A.):
   A vertical asymptote occurs when the denominator of the function is zero, because the function approaches infinity near these points.

   For the function $f(x) = \frac{1}{x - 1}$, the vertical asymptote occurs where $x - 1 = 0$, which simplifies to: $x = 1$ Therefore, the equation for the vertical asymptote is: $\text{Vertical Asymptote: } x = 1$

2. Horizontal Asymptote (H.A.):
   A horizontal asymptote occurs when $x$ approaches infinity or negative infinity, and we look at the behavior of the function.

   For rational functions like $f(x) = \frac{1}{x - 1}$, as $x$ tends to $\infty$ or $-\infty$, the function approaches zero because the numerator is a constant (1) and the denominator grows without bound.

   Thus, the horizontal asymptote is: $\text{Horizontal Asymptote: } y = 0$

Explanation of the error messages:

• For the Horizontal Asymptote, you wrote x = 1, which is incorrect because that is the vertical asymptote equation, not the horizontal one. The correct horizontal asymptote is $y = 0$.

• For the Vertical Asymptote, you wrote y = 0, which is also incorrect because the vertical asymptote relates to the value of $x$, not $y$. The correct vertical asymptote is $x = 1$.

Corrected Asymptote Equations:

• Horizontal Asymptote: $y = 0$
• Vertical Asymptote: $x = 1$

Do you want further explanation on asymptotes or how to graph them?

Here are 5 questions that expand on this:

1. How do horizontal asymptotes change when the degree of the numerator is higher than the denominator?
2. Can a function cross its horizontal asymptote, and if so, when?
3. How do we find oblique (slant) asymptotes?
4. How does the graph of $f(x) = \frac{1}{x - a}$ change as $a$ varies?
5. What happens to the asymptotes if the numerator is a higher-degree polynomial?

Tip: Always check the degrees of the numerator and denominator when identifying horizontal asymptotes for rational functions.