Question

question 39
the graph of a rational function, f, is shown here with all of its asymptotes. find the domain and range of f.
(graph of the rational function with asymptotes and grid, along with multiple - choice options for domain and range:

• domain $(-\infty,1)\cup(1,\infty)$

range $(-\infty,3)\cup(3,\infty)$

• domain $(-10,10)$

range $(-10,10)$

• domain $(-\infty,3)\cup(3,\infty)$

range $(-\infty,1)\cup(1,\infty)$

• domain $(-\infty,3)\cup(3,\infty)$

range $(-\infty,\infty)$)

Explanation:

Step1: Identify vertical asymptote

The vertical dashed line is at $x=3$, so $x=3$ is excluded from the domain. Thus, domain is $(-\infty,3)\cup(3,\infty)$.

Step2: Identify horizontal asymptote

The horizontal dashed line is at $y=1$, so $y=1$ is excluded from the range. The graph has values below 1 and above 1, so range is $(-\infty,1)\cup(1,\infty)$.

Step3: Match to options

Compare with given choices to find the correct pair.

Answer:

Domain: $(-\infty,3)\cup(3,\infty)$
Range: $(-\infty,1)\cup(1,\infty)$
(Option: Domain $(-\infty,3)\cup(3,\infty)$; Range $(-\infty,1)\cup(1,\infty)$)