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. Identify the horizontal asymptote by observing the behavior of the graph as x approaches positive and negative infinity. The graph approaches y = -2, indicating a horizontal asymptote at y = -2.
2. Check for vertical asymptotes by looking for values of x where the graph does not exist or approaches infinity. In this graph, there are no vertical asymptotes as the graph is continuous.
3. Determine if there are any oblique asymptotes by checking if the graph approaches a slanted line as x approaches infinity. In this case, there are no oblique asymptotes.
4. State the domain of the function by identifying all x-values for which the function is defined. Since the graph is continuous and there are no breaks, the domain is all real numbers, (-∞, ∞).
5. Summarize the findings: The function has a horizontal asymptote at y = -2, no vertical or oblique asymptotes, and a domain of all real numbers.

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where a function approaches infinity, typically at values that make the denominator zero. Horizontal asymptotes indicate the behavior of a function as x approaches infinity or negative infinity, showing the value the function approaches. Oblique asymptotes occur when the degree of the numerator is one higher than that of the denominator.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero, as these lead to undefined outputs. Understanding the domain is crucial for identifying vertical asymptotes and ensuring the function's behavior is accurately represented.

Graph Behavior at Infinity

Graph behavior at infinity refers to how a function behaves as the input values grow very large or very small. This concept is essential for determining horizontal asymptotes, which indicate the value the function approaches as x approaches positive or negative infinity. Analyzing this behavior helps in understanding the long-term trends of the function's graph.