Question

lf f : [1, ∞) `rightarrow` [2, ∞) is given by f(x) = `x + 1/x`, then f^–1(x) is equal to ______.

विकल्प

• `(x + sqrt(x^2 - 4))/2`

• `x/(1 + x^2)`

• `(x - sqrt(x^2 - 4))/2`

• `1 + sqrt(x^2 - 4)`

Answer 1

lf f : [1, ∞) `rightarrow` [2, ∞) is given by f(x) = `x + 1/x`, then f^–1(x) is equal to `underlinebb((x + sqrt(x^2 - 4))/2)`.

Explanation:

Let y = `x + 1/x`

`\implies` y = `(x^2 + 1)/x`

`\implies` xy = x² + 1

`\implies` x² – xy + 1 = 0

`\implies` x = `(y ± sqrt(y^2 - 4))/2`

`\implies` f^–1(y) = `(y ± sqrt(y^2 - 4))/2`

∴  f^–1(x) = `(x ± sqrt(x^2 - 4))/2`

Since, the range of the inverse function is [1, ∞), then we take f^–1(x) = `(x + sqrt(x^2 - 4))/2`

If we consider, f^–1(x) = `(x - sqrt(x^2 - 4))/2`, then f^–1(x) > 1.

This is possible only, when (x – 2)² > x² – 4

`\implies` x² + 4 – 4x > x² – 4

`\implies` 8 > 4x

`\implies` x < 2, when x > 2