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

Verified step by step guidance

1. 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).
2. 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 \( x = a \).
3. Step 3: Determine horizontal asymptotes by analyzing the end behavior of the function as \( x \to \infty \) and \( x \to -\infty \). Calculate \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \). If these limits approach a finite number \( L \), then \( y = L \) is a horizontal asymptote.
4. 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).
5. 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.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value, often where the denominator of a rational function is zero. They represent values excluded from the domain and indicate where the graph has a vertical line that the curve approaches but never touches.

Horizontal and Oblique Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches positive or negative infinity, showing the value the function approaches. Oblique (slant) asymptotes occur when the function approaches a non-horizontal line, typically found by polynomial division when the numerator’s degree is exactly one more than the denominator’s.

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero. Identifying the domain is essential to understanding where the function exists and where asymptotes may occur.