For R(x) = (x^2 - 4)/(x^2 - 1), identify the domain and any holes or vertical asymptotes.

For R(x) = (x^2 - 4)/(x^2 - 1), identify the domain and any holes or vertical asymptotes.

• A. Domain: all real numbers except x=±1; vertical asymptotes at x=±1; no holes.
• B. Domain: all real numbers; vertical asymptotes at x=±1; no holes.
• C. Domain: all real numbers except x=±1; vertical asymptotes at x=±1; holes at x=±1.
• D. Domain: all real numbers except x=±1; vertical asymptotes at x=±1; a hole at x=±1.

Answer: (A)

The main idea is that a hole happens only when you can cancel a common factor between the numerator and denominator, creating a removable discontinuity. Here, x^2-4 factors as (x-2)(x+2) and x^2-1 factors as (x-1)(x+1); there are no shared factors, so nothing cancels. The domain is thus all real numbers except where the denominator is zero, which is x=±1. Since the numerator is nonzero at those points (x^2-4 = -3 when x=±1), those points are vertical asymptotes, not holes. So the domain is all real numbers except ±1; vertical asymptotes at ±1; no holes.

The main idea is that a hole happens only when you can cancel a common factor between the numerator and denominator, creating a removable discontinuity. Here, x^2-4 factors as (x-2)(x+2) and x^2-1 factors as (x-1)(x+1); there are no shared factors, so nothing cancels. The domain is thus all real numbers except where the denominator is zero, which is x=±1. Since the numerator is nonzero at those points (x^2-4 = -3 when x=±1), those points are vertical asymptotes, not holes. So the domain is all real numbers except ±1; vertical asymptotes at ±1; no holes.