Basic Concepts 5.2 Quiz Question 8 of 10 This question: 1 point(s) possible Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational function. State the domain of f. \[ f(x)=\frac{2}{x-4} \] Identify any vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is one vertical asymptote. Its equation is \( \square \) . (Type an equation.) B. There are two vertical asymptotes. The equation of the leftmost one is \( \square \) and the equation of the rightmost one is \( \square \) . (Type equations.) C. There are no vertical asymptotes. Identify the horizontal or oblique asymptote, if one exists. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function has a horizontal asymptote whose equation is \( \square \) . (Type an equation.) B. The function has an oblique asymptote whose equation is \( \square \) . (Type an equation.) C. The function has neither a horizontal nor an oblique asymptote. The domain of \( f \) is \( \square \) .

Let's analyze the function \( f(x) = \frac{2}{x - 4} \) to determine its vertical and horizontal asymptotes, as well as its domain. ### 1. **Vertical Asymptotes** Vertical asymptotes occur where the denominator of the rational function is zero (and the numerator is not zero at those points). - **Set the denominator equal to zero:** \[ x - 4 = 0 \implies x = 4 \] - **Conclusion:** There is **one vertical asymptote** at \( x = 4 \). **Choice A:** *A. There is one vertical asymptote. Its equation is \( x = 4 \).* ### 2. **Horizontal or Oblique Asymptotes** To find horizontal or oblique asymptotes, compare the degrees of the numerator and the denominator. - **Degrees:** - Numerator (\(2\)) has degree **0**. - Denominator (\(x - 4\)) has degree **1**. - **Since the degree of the numerator is less than the degree of the denominator,** the horizontal asymptote is \( y = 0 \). - **No oblique asymptote exists** in this case. **Choice A:** *A. The function has a horizontal asymptote whose equation is \( y = 0 \).* ### 3. **Domain of \( f \)** The domain consists of all real numbers except where the denominator is zero. - **Excluded value:** \( x = 4 \) - **Domain in interval notation:** \[ (-\infty, 4) \cup (4, \infty) \] ### **Summary of Answers** - **Vertical Asymptote:** \( x = 4 \) - **Horizontal Asymptote:** \( y = 0 \) - **Domain of \( f \):** \( (-\infty, 4) \cup (4, \infty) \)