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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.49
Chapter 2, Problem 2.49

Horizontal and Vertical Asymptotes


Determine the domain and range of y = (√16―x²) / (x―2).

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Step 1: Identify the domain of the function. The domain is determined by the values of x for which the function is defined. The function y = (√(16 - x²)) / (x - 2) has two restrictions: the expression under the square root, 16 - x², must be non-negative, and the denominator, x - 2, must not be zero.
Step 2: Solve the inequality 16 - x² ≥ 0 to find the values of x for which the square root is defined. This inequality can be rewritten as x² ≤ 16, which implies -4 ≤ x ≤ 4.
Step 3: Determine the values of x that make the denominator zero. Set x - 2 = 0 and solve for x, which gives x = 2. Since the denominator cannot be zero, x = 2 is excluded from the domain.
Step 4: Combine the results from Steps 2 and 3 to find the domain. The domain is the set of all x such that -4 ≤ x ≤ 4, excluding x = 2. In interval notation, this is [-4, 2) ∪ (2, 4].
Step 5: Determine the range of the function. Consider the behavior of the function as x approaches the endpoints of the domain and the point where the denominator is zero. Analyze the limits as x approaches 2 from the left and right to identify any vertical asymptotes, and consider the maximum and minimum values of the function within the domain to determine the range.

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Key Concepts

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

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero. In this case, we need to identify any x-values that would cause the denominator (x - 2) to equal zero, as these values are excluded from the domain.
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Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range, we analyze the behavior of the function as x approaches certain critical points, including vertical asymptotes and the limits of the function as x approaches infinity or negative infinity. This helps in understanding the values that y can take.
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Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where the function is undefined, typically where the denominator is zero. Horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity, indicating the value that y approaches. Understanding these concepts is crucial for analyzing the overall behavior of the function.
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