Find the domain and the vertical and horizontal asymptotes. Separate your asymptote answers with commas, if there are multiple, or enter "DNE" if there are none. \( f(x)=\frac{-1}{x-4} \) Domain: Vertical asymptote(s): \( x= \) Horizontal asymptote(s): \( y= \)

To analyze the function \( f(x) = \frac{-1}{x-4} \), we will determine the domain, vertical asymptotes, and horizontal asymptotes. ### Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The function \( f(x) \) is undefined when the denominator is zero. Set the denominator equal to zero to find the restriction: \[ x - 4 = 0 \implies x = 4 \] Thus, the function is undefined at \( x = 4 \). Therefore, the domain is all real numbers except \( x = 4 \): \[ \text{Domain: } (-\infty, 4) \cup (4, \infty) \] ### Vertical Asymptote: Vertical asymptotes occur where the function approaches infinity or negative infinity, typically where the denominator is zero (and the numerator is not zero). Since we found that the function is undefined at \( x = 4 \), there is a vertical asymptote at: \[ \text{Vertical asymptote: } x = 4 \] ### Horizontal Asymptote: Horizontal asymptotes describe the behavior of the function as \( x \) approaches infinity or negative infinity. For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \). In this case, the numerator is a constant (degree 0) and the denominator is linear (degree 1). Therefore, as \( x \) approaches infinity or negative infinity, \( f(x) \) approaches 0: \[ \text{Horizontal asymptote: } y = 0 \] ### Summary: - **Domain:** \( (-\infty, 4) \cup (4, \infty) \) - **Vertical asymptote(s):** \( x = 4 \) - **Horizontal asymptote(s):** \( y = 0 \) So, the final answers are: - Domain: \( (-\infty, 4) \cup (4, \infty) \) - Vertical asymptote(s): \( x = 4 \) - Horizontal asymptote(s): \( y = 0 \)