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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 57
Chapter 4, Problem 57

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

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Step 1: Identify the vertical asymptotes by looking for values of x where the function approaches infinity or negative infinity. These typically occur where the denominator of a rational function is zero. From the graph, observe if the curve approaches a vertical line but never touches it.
Step 2: Identify the horizontal asymptotes by observing the behavior of the function as x approaches positive or negative infinity. The graph shows the function approaching a horizontal line, which indicates a horizontal asymptote. Note the y-value of this line.
Step 3: Check for oblique (slant) asymptotes by examining if the function approaches a non-horizontal, non-vertical line as x approaches infinity or negative infinity. In this graph, the function does not approach a slant line, so no oblique asymptote is present.
Step 4: State the domain of the function by excluding any x-values where vertical asymptotes occur, since the function is undefined at those points. The domain includes all real numbers except these x-values.
Step 5: Summarize the asymptotes and domain: vertical asymptotes correspond to x-values where the function is undefined, the horizontal asymptote corresponds to the y-value the function approaches at infinity, and the domain excludes the vertical asymptote points.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 appear as vertical lines on the graph.
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Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches positive or negative infinity. They are horizontal lines that the graph approaches but does not necessarily touch, indicating the end behavior of the function.
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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, which often correspond to vertical asymptotes.
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