Inverse Functions

A function admits an inverse on its range when it is one-to-one, and the inverse is found by solving y = f(x) for x, a standard JEE Advanced algebraic procedure for Mobius-type maps. Set y = (3x - 2)/(x + 4), so y(x + 4) = 3x - 2. Expanding gives yx + 4y = 3x - 2, then group x terms: yx - 3x = -2 - 4y, so x(y - 3) = -(2 + 4y). Therefore x = -(2 + 4y)/(y - 3) = (4y + 2)/(3 - y). Option (2 - 4y)/(y - 3) has a sign error in the numerator. Option (4y + 2)/(y - 3) keeps the wrong denominator sign. Option (3y - 2)/(y + 4) merely renames the original function rather than inverting it. Hence f inverse of y equals (4y + 2)/(3 - y). Plausibility check: substituting y = f(0) = -1/2 should return x = 0, and (4(-1/2)+2)/(3+1/2) = 0/3.5 = 0, confirming the inverse expression is correct. Fractional linear maps of this kind form a group under composition, and the inverse of such a map is again fractional linear, a structural fact worth carrying into the exam hall. Verifying the candidate inverse on one convenient input value, as done above, remains the single fastest safeguard against the sign and grouping slips that derail these algebraic inversions.