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 asymptote by observing where the graph approaches a vertical line but never crosses it. In the graph, the vertical asymptote is at \(x = 7\), indicated by the dashed blue vertical line.
2. Step 2: Identify the horizontal asymptote by looking for a horizontal line that the graph approaches as \(x\) goes to positive or negative infinity. The graph approaches the horizontal line \(y = 6\) (green dashed line) as \(x\) goes to both positive and negative infinity.
3. Step 3: Check for any oblique (slant) asymptotes by seeing if the graph approaches a non-horizontal, non-vertical line as \(x\) goes to infinity. In this graph, there is no oblique asymptote since the graph approaches a horizontal line instead.
4. Step 4: State the domain of the function \(f\). Since there is a vertical asymptote at \(x = 7\), the function is undefined at this point. Therefore, the domain is all real numbers except \(x = 7\), which can be written as \((-\infty, 7) \cup (7, \infty)\).
5. Step 5: Summarize the asymptotes and domain: Vertical asymptote at \(x = 7\), horizontal asymptote at \(y = 6\), no oblique asymptote, and domain \((-\infty, 7) \cup (7, \infty)\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value. They often correspond to values that make the denominator of a rational function zero, indicating points where the function is undefined.

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 asymptotes occur when the function approaches a slanted line, typically when the degree of the numerator is one more than the denominator in a rational function.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. For rational functions, the domain excludes values that cause division by zero, often corresponding to vertical asymptotes.