Question

determine the domain and the equations of the asymptotes for the graph of the rational function.
the domain is (-∞,∞). (simplify your answer. type your answer in interval notation.)

Explanation:

Step1: Recall domain of rational function

The domain of a rational function $y = \frac{f(x)}{g(x)}$ is all real - numbers except the values of $x$ that make the denominator $g(x)=0$. Since no function is given explicitly, if we assume a non - zero denominator for all real $x$, the domain is all real numbers. In interval notation, it is $(-\infty,\infty)$.

Step2: Recall vertical asymptote rule

Vertical asymptotes occur at the values of $x$ that make the denominator of a rational function equal to zero. Since the denominator is non - zero for all real $x$ (as per our domain assumption), there are no vertical asymptotes.

Step3: Recall horizontal asymptote rule

For a rational function $\frac{f(x)}{g(x)}$ where $\text{deg}(f(x)) = m$ and $\text{deg}(g(x))=n$. If $m < n$, the horizontal asymptote is $y = 0$. Without the specific function, if we assume a general well - behaved rational function, we consider cases. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.

Answer:

Domain: $(-\infty,\infty)$; No vertical asymptotes; Horizontal asymptote: $y = 0$