Final answer:
The function f(x) = x^2 / (x^2 - 1) is a rational function with a domain of (-∞, -1) U (-1, 1) U (1, ∞), vertical asymptotes at x = -1 and x = 1, a horizontal asymptote at y = 1, and an x-intercept at x = 0.
Step-by-step explanation:
The function f(x) = x^2 / (x^2 - 1) is a rational function. In order to analyze its behavior, we need to determine its domain, vertical asymptotes, horizontal asymptotes, and x-intercepts.
The function is defined for all real numbers except when the denominator is equal to zero. So we need to find the values of x that make x^2 - 1 = 0. Solving the equation, we get x = 1 and x = -1.
Therefore, the domain of the function is (-∞, -1) U (-1, 1) U (1, ∞). The vertical asymptotes occur when the denominator is equal to zero, so the vertical asymptotes are x = -1 and x = 1. Since the degree of the numerator and denominator are the same, the horizontal asymptote can be found by dividing the leading coefficients, which gives us y = 1. Finally, to find the x-intercepts, we set the numerator equal to zero, which gives us x = 0.