f is the rational function if f(x) = N(x)/D(x) for some polynomials N(x) and D(x). Fraction N(x)/D(x) is proper if deg(N(x)) < deg(D(x)). Otherwise, it is improper. As value of the proper fraction approaches 0 as x approaches either positive or negative infinity, x-axis is both left and right horizontal asymptote for f is fraction is proper. Otherwise, by division, there is the quotient polynomial Q(x) and remainder polynomial R(x) such that f(x) = Q(x) + R(x)/D(x) where either R(x) = 0 or deg(R(x)) < deg(D(x)). So, in any case, both left and right asymptote for graph of f is y = Q(x). As deg(Q(x)) = deg(N(x)) - deg(D(x)) when N(x)/D(x) is improper, left-right asymptote for rational function may be polynomial curve rather than line. Determine left-right asymptotes for given functions.

i. f(x) = (2x)/(x2 - 1).

ii. f(x) = (3x - 2)/(x + 3).

iii. f(x) = (x2 - x + 2)/(2x + 1).

iv. f(x) = (2x3 - x + 2)/(2x - 1).

Also, range of rational function, as any function, is discovered analytically by solving function formula for x in terms of y. If both N(x) and D(x) have degrees at most 2, then quadratic formula may be used to determine range of  rational function. Determine range of given functions.

i. f(x) = (x + 1)/(2x - 3).

ii. f(x) = (4x)/(x2 + 1).

iii. f(x) = (x2 + x - 1)/(2x - 1).