Final answer:
The function f(x)=(x²-2)/(x-1) has a vertical asymptote at x=1 and a horizontal asymptote at y=1. The domain is all real numbers except x=1, and the range is all real numbers.
Step-by-step explanation:
To find the vertical and horizontal asymptotes of the function f(x)=(x²-2)/(x-1), we first simplify the function if possible and then look at the behavior as x approaches certain critical values. For vertical asymptotes, we investigate values of x that make the denominator zero. Here, the denominator is zero when x is 1, so there is a vertical asymptote at x = 1.
The horizontal asymptote is determined by looking at the behavior of the function as x approaches infinity. Since the degree of the numerator and the denominator is the same, we take the ratio of the leading coefficients, which is 1. Thus, the horizontal asymptote is y = 1.
The domain of a function is the set of all possible x values. Given that the denominator cannot be zero, the domain of f(x) is all real numbers except x = 1. So the domain is x ∈ (-∞, 1) ∪ (1, +∞).
The range of the function is all the y values that f(x) can take. Since we have a horizontal asymptote at y = 1 and vertical asymptote at x = 1, but no other restrictions on the y values, the range is all real numbers. Thus the range is y ∈ (-∞, +∞).