Range Of Functions

Finding the range of a rational function is often done by setting y = g(x) and requiring the resulting quadratic in x to have real solutions, a standard JEE Advanced discriminant method. Write y = (x^2 + x + 1)/(x^2 + x + 2). Let t = x^2 + x, whose minimum is -1/4 since x^2 + x = (x + 1/2)^2 - 1/4, so t ranges over [-1/4, ∞). Then g = (t + 1)/(t + 2) = 1 - 1/(t + 2). As t increases from -1/4 to ∞, t + 2 ranges over [7/4, ∞), so 1/(t + 2) ranges over (0, 4/7], making g range over [1 - 4/7, 1) = [3/7, 1). Option (0, 1] mishandles both endpoints. Option [1/2, 1) uses the wrong minimum of t. Option (1, ∞) ignores that g is always below 1. Hence the range is [3/7, 1). Plausibility check: the minimum 3/7 occurs at x = -1/2 where t = -1/4, and g never reaches 1 since the numerator is always less than the denominator, confirming the half-open interval.