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

Verified step by step guidance

1. Step 1: Identify the vertical asymptotes by looking for values of x where the function approaches infinity or negative infinity. In the graph, these are the vertical dashed lines where the function is undefined. Here, the vertical asymptotes are at \(x = -4\) (red dashed line) and \(x = 3\) (orange dashed line).
2. Step 2: Identify the horizontal asymptote by observing the behavior of the function as \(x\) approaches positive or negative infinity. The graph shows the function approaching the horizontal dashed purple line \(y = -2\) as \(x\) goes to both \(+\infty\) and \(-\infty\). So, the horizontal asymptote is \(y = -2\).
3. Step 3: Check for any oblique (slant) asymptotes by looking for a line that the graph approaches as \(x\) goes to infinity, which is neither horizontal nor vertical. In this graph, there is no oblique asymptote since the function approaches a horizontal line instead.
4. Step 4: State the domain of the function \(f\). The domain includes all real numbers except where the vertical asymptotes occur, because the function is undefined at those points. Therefore, the domain is all real numbers except \(x = -4\) and \(x = 3\).
5. Step 5: Summarize the asymptotes and domain: Vertical asymptotes at \(x = -4\) and \(x = 3\), horizontal asymptote at \(y = -2\), no oblique asymptote, and domain \((-\infty, -4) \cup (-4, 3) \cup (3, \infty)\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value. These are typically found where the denominator of a rational function is zero, causing the function to be undefined. In the graph, vertical asymptotes are shown as vertical dashed lines where the curve shoots up or down without bound.

Horizontal and Oblique Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity, indicating the value the function approaches. Oblique (slant) asymptotes occur when the function approaches a line with a non-zero slope at infinity. In the graph, the horizontal asymptote is a horizontal dashed line, while an oblique asymptote would be a slanted dashed line.

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 cause division by zero, often corresponding to vertical asymptotes. Identifying vertical asymptotes helps determine the domain by excluding those x-values.