Textbook Question
Graph each rational function. See Examples 5–9. ƒ(x)=(x+2)/(x-3)
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5. Rational Functions
Graphing Rational Functions
2:40 minutesProblem 58
Textbook Question
Identify any vertical, horizontal, or oblique asymptotes in the graph of . State the domain of .
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Verified step by step guidance1
To identify vertical asymptotes, look for values of x that make the denominator of the function zero, as these are points where the function is undefined. Solve the equation set by the denominator equal to zero.
For horizontal asymptotes, compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, the horizontal asymptote is the ratio of the leading coefficients.
If the degree of the numerator is exactly one more than the degree of the denominator, there is an oblique (or slant) asymptote. Perform polynomial long division to find the equation of the oblique asymptote.
To determine the domain of the function, identify all x-values that make the denominator zero, as these are excluded from the domain. The domain is all real numbers except these values.
Summarize the findings: list the vertical asymptotes, horizontal or oblique asymptotes, and state the domain of the function based on the excluded x-values.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Asymptotes
Asymptotes are lines that a graph approaches but never touches. They can be vertical, horizontal, or oblique (slant). Vertical asymptotes occur where a function approaches infinity, typically at points where the denominator of a rational function is zero. Horizontal asymptotes indicate the behavior of a function as x approaches infinity or negative infinity, while oblique asymptotes describe linear behavior when the degree of the numerator is one higher than that of the denominator.
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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. Understanding the domain is crucial for identifying vertical asymptotes, as these occur at values where the function is undefined, typically due to division by zero. To determine the domain, one must analyze the function's formula and identify any restrictions that would prevent certain x-values from being valid inputs.
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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 identifying horizontal and oblique asymptotes. By evaluating the limits of the function as x approaches positive or negative infinity, one can determine if the function stabilizes at a particular value (horizontal asymptote) or follows a linear trend (oblique asymptote). Understanding this behavior helps in sketching the overall shape of the graph.
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