If f:[1,∞) → [2,∞) is given by f(x) = x + 1/x, then f^-1(x) equals

If f:[1,∞) → [2,∞) is given by f(x) = x + 1/x, then f^-1(x)… — JEE Main 2001 Mathematics PYQ

2001

f:[1,\infty) \to [2,\infty)

is given by

f(x) = x + \frac{1}{x}

, then

f^{-1}(x)

equals

A.
$\frac{x + \sqrt{x^2 - 4}}{2}$
B.
$\frac{x}{1 + x^2}$
C.
$\frac{x - \sqrt{x^2 - 4}}{2}$
D.
$1 + \sqrt{x^2 - 4}$

Show answer and solution

Correct option: A

Solution

To find f⁻¹(x), set y = x + 1/x, so x² - yx + 1 = 0. Solving: x = (y ± √(y² - 4))/2. Since x ∈ [1,∞), we need the larger root, giving f⁻¹(x) = (x + √(x²-4))/2. The minus sign would give values in (0,1], which is outside the domain.

[2001]

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