Algebra 2 Honors Practice Test

For f(x) = (x^2 - 1)/(x^2 - 4), identify the domain and vertical/horizontal asymptotes.

Domain ℝ; vertical asymptotes x = ±2; horizontal asymptote y = 1 Domain ℝ \ {±2}; vertical asymptotes x = ±2; horizontal asymptote y = 1 Domain ℝ \ {±2}; vertical asymptotes x = 0; horizontal asymptote y = 0 Domain ℝ \ {±3}; vertical asymptotes x = ±3; horizontal asymptote y = 1

Correct Answer:

Domain ℝ \ {±2}; vertical asymptotes x = ±2; horizontal asymptote y = 1

Explanation:

For a rational function, the domain excludes points where the denominator is zero. Here the denominator x^2 - 4 factors to (x - 2)(x + 2), which is zero at x = ±2, so all real numbers except ±2 are allowed. At those x-values the numerator x^2 - 1 equals 3, not zero, so the function shoots to infinity there, giving vertical asymptotes at x = ±2. Since both numerator and denominator are degree 2, the end behavior is governed by the ratio of leading coefficients, which is 1, so the horizontal asymptote is y = 1. There are no common factors, so no holes occur. Domain: all real numbers except ±2 Vertical asymptotes: x = ±2 Horizontal asymptote: y = 1

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