If the function f:[1,∞) → [1,∞) is defined by f(x) = 2^x(x-1), then f^-1(x) is

If the function f:[1,∞) → [1,∞) is defined by f(x) =… — JEE Main 1999 Mathematics PYQ

1999

If the function

f:[1,\infty) \to [1,\infty)

is defined by

f(x) = 2^{x(x-1)}

, then

f^{-1}(x)

A.
$\left(\frac{1}{2}\right)^{x(x-1)}$
B.
$\frac{1}{2}(1+\sqrt{1+4\log_2 x})$
C.
$\frac{1}{2}(1-\sqrt{1+4\log_2 x})$
D.
not defined

Show answer and solution

Correct option: B

Solution

Set y = 2^(x(x-1)), so log₂y = x(x-1) = x²-x. This gives x²-x-log₂y = 0. Solving via quadratic formula: x = (1 ± √(1+4log₂y))/2. Since x ≥ 1, we take the positive root, giving f⁻¹(x) = (1+√(1+4log₂x))/2.

[1999]

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