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, assume the general case of avoiding division by zero.

Step2: Consider vertical asymptotes

Vertical asymptotes occur at the values of $x$ that make the denominator of the rational function equal to zero. Let the rational function be $y=\frac{f(x)}{g(x)}$, vertical asymptotes are at $x = a$ where $g(a)=0$ and $f(a)
eq0$.

Step3: Consider horizontal asymptotes

If the degree of the numerator $n$ and the degree of the denominator $m$ of a rational function $y=\frac{f(x)}{g(x)}$:

• If $n\lt m$, the horizontal asymptote is $y = 0$.
• If $n=m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$, where $a_n$ and $b_m$ are the leading coefficients of $f(x)$ and $g(x)$ respectively.
• If $n\gt m$, there is no horizontal asymptote.

Since no specific rational function is given, for the domain, assume no denominator information. The domain of a rational function in the most general non - restricted case (if we assume the denominator is non - zero everywhere possible) is $(-\infty,\infty)$ with possible exclusions of $x$ values that make the denominator zero. But without the function, we assume no such values for now.

Answer:

Domain: $(-\infty,\infty)$
(Note: If a specific rational function was provided, we would need to find the values of $x$ that make the denominator zero and exclude them from the domain. Also, we would need to analyze the numerator and denominator degrees to find the equations of the asymptotes accurately.)