If f : [ 1 , ) [ 2 , ) is given by f ( x ) = x + 1 x , then f - 1 ( x ) equals [2001]

Question

If $f : [1, \infty) \to [2, \infty)$ is given by $f (x) = x + \frac{1}{x},$ then $f^{- 1} (x)$ equals [2001]

Smart Achievers · Accepted Answer

(1)

$Given: f (x) = x + \frac{1}{x} = y$

$(let) \Rightarrow x^{2} - y x + 1 = 0$

$\Rightarrow x = \frac{y \pm \sqrt{y^{2} - 4}}{2}$

$∴$ $x = \frac{y + \sqrt{y^{2} - 4}}{2}$ $[∵ x \geq 1 and y \geq 2]$

$∴$ $f^{- 1} (x) = \frac{x + \sqrt{x^{2} - 4}}{2}$

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