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

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.