Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=f(x)y=f\left(x\right). State the domain of ff.

Accepted Answer

Step 1: Identify the function $$ f(x) $$ from the given graph or expression. Since the problem references a graph, observe the behavior of the function near points where it might be undefined or where the graph shows unusual behavior (like breaks or infinite limits). Step 2: Find vertical asymptotes by locating values of $$ x $$ where the function is undefined and the limit of $$ f(x) $$ approaches infinity or negative infinity. Typically, these occur where the denominator of a rational function is zero (if applicable). Write these values as vertical asymptotes $$ x = a . $$Step 3: Determine horizontal asymptotes by analyzing the end behavior of the function as $$ x 	o \infty $$ and $$ x 	o -\infty . $$Calculate $$ \lim_{x 	o \infty} f(x) $$ and $$ \lim_{x 	o -\infty} f(x) . If $$these limits approach a finite number $$ L $$, then $$ y = L  is a $$horizontal asymptote. Step 4: Check for oblique (slant) asymptotes if the degree of the numerator is exactly one more than the degree of the denominator (for rational functions). Perform polynomial long division of the numerator by the denominator to find the quotient $$ q(x) . $$The oblique asymptote is given by $$ y = q(x)  (a $$linear function). Step 5: State the domain of $$ f  by $$excluding any values of $$ x $$ that cause the function to be undefined (such as points where the denominator is zero or where the function has vertical asymptotes). Express the domain in interval notation.

Identify any vertical, horizontal, or oblique asymptotes in the graph of $y = f (x)$ . State the domain of $f$ .

Verified video answer for a similar problem:

Key Concepts

Vertical Asymptotes

Horizontal and Oblique Asymptotes

Domain of a Function

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=f(x)y=f\(\left\)(x\(\right\)). State the domain of ff.