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Have to write a equation for the vertical and horizontal adymotote cannot just be “x=1 or y=0”

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.

Have to write an equation for the vertical and horizontal asymptote, cannot just be 'x = 1' or 'y = 0'.

Learn how to determine the vertical and horizontal asymptotes for the rational function f(x) = 1/(x-1). This problem involves finding the vertical asymptote by setting the denominator to zero and understanding the behavior of the function as x approaches infinity for the horizontal asymptote. Suitable for high school students in Grades 10-12.

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