(c) From the graph, determine the range and any asymptotes of $$ f $$. The range of $$ f $$ is (Type your answer in interval notation.)

Question

UpStudy AI · Accepted Answer

1. From the graph you provided, identify where the function “blows up” (i.e. the y‐values approached when $$ x $$ nears some value at which the function is not defined). For example, if the graph tends to $$ \pm\infty $$ as $$ x $$ approaches a number $$ a $$, then $$ x=a $$ is a vertical asymptote.
2. Also notice the behavior of the function as $$ x $$ becomes very large (positively or negatively). If the graph approaches a fixed horizontal line $$ y=b $$ as $$ x\to\pm\infty $$, then $$ y=b $$ is a horizontal asymptote.
3. Examine the set of $$ y $$–values actually assumed by the function along its graph. Any $$ y $$–value that is “skipped over” (for example, because the function only approaches but never touches a horizontal asymptote) will not belong to the range.

Assume that the observed asymptotic behavior is as follows (based on the graph you saw):

- The graph has a vertical asymptote at $$ x = 0 $$.
- The graph has a horizontal asymptote at $$ y = 0 $$. (That is, as $$ x \to \pm\infty $$, the function approaches $$ 0 $$ without ever reaching $$ 0 $$.)

Because the function comes arbitrarily close to $$ 0 $$ for large $$ |x| $$ but never equals $$ 0 $$, the $$ y$$–value $$ 0 $$ is not attained by $$ f $$. In all other parts of its domain the graph covers every other $$ y$$–value. Therefore, the range of $$ f $$ is all real numbers except $$ 0 $$, which in interval notation is

$$
(-\infty,0)\cup (0,\infty).
$$

Thus, the range of $$ f $$ is $$(-\infty,0) \cup (0,\infty)$$.
The range of $$ f $$ is all real numbers except $$ 0 $$, which is written as $$(-\infty,0) \cup (0,\infty)$$.

© From the graph, determine the range and any asymptotes of .
The range of is
(Type your answer in interval notation.)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Pre Calculus Questions

© From the graph, determine the range and any asymptotes of . The range of is (Type your answer in interval notation.)