JEE Main 2021 · 18 March · evening · Single correct · +4 / −1

Let f : R - 3 → R - 1 be defined by f(x) = (x - 2)/(x - 3).
Let g : R → R be given as g(x) = 2x - 3. Then, the sum of all the values of x for which f⁻¹(x) + g⁻¹(x) = 13/2 is equal to :

  1. (a) 3
  2. (b) 5
  3. (c) 2
  4. (d) 7
Answer and solution

Answer (b) 5

Solution

Maths sir · 5 steps

Find both inverses first, then the equation becomes a plain quadratic in x.

  1. Invert f. Put y = (x-2)/(x-3) and solve for x. You get y(x-3) = x-2, so x(y-1) = 3y-2. This gives f⁻¹(x) = (3x-2)/(x-1), defined for x ≠ 1.
  2. Invert g. From y = 2x-3 you get x = (y+3)/2, so g⁻¹(x) = (x+3)/2.

    (3x-2)/(x-1) + (x+3)/2 = 13/2

  3. Clear fractions. Multiply by 2(x-1): 2(3x-2) + (x+3)(x-1) = 13(x-1). That is x² + 8x - 7 = 13x - 13, so x² - 5x + 6 = 0.
  4. Check the roots. x² - 5x + 6 = (x-2)(x-3) = 0 gives x = 2 and x = 3. Neither equals 1, so both lie in the domain of f⁻¹.
  5. So. The sum of all such values is 2 + 3 = 5, which is option (b).

The idea to keep. To invert a function, write y = f(x) and solve for x. After clearing fractions, always check that each root lies in the domain of the inverse.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Functions (Advanced)

Asked about 70 times in JEE Main since 2020. This question is from 2021.