Horizontal and Vertical Asymptotes


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

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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].
5. 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.

Key Concepts

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.

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.

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.