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Algebra Level 3

If f ( x ) = x ∣ x ∣ f(x) = x|x| , then f − 1 ( x ) f^{-1}(x) is equal to?

Clarification:

sgn ( x ) \text{sgn}(x) is the signum function defined as sgn ( x ) = { 1 , x > 0 0 , x = 0 − 1 , x < 0 \text{sgn}(x)=\begin{cases} 1 \quad , x >0 \\ 0 \quad , x=0 \\ -1 \ \ , x<0 \end{cases} .

∣ x ∣ \sqrt{\mid x\mid} sgn ( x ) ∣ x ∣ \text{sgn}(x) \sqrt{\mid x\mid} None of these choices − ∣ x ∣ -\sqrt{\mid x\mid}

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Chew-Seong Cheong
Sep 22, 2015

f ( x ) = x ∣ x ∣ = { − ∣ x 2 ∣ for x < 0 0 for x = 0 ∣ x 2 ∣ for x > 0 = sgn ( x ) ∣ x 2 ∣ ⇒ f − 1 ( x ) = sgn ( x ) ∣ x ∣ f(x) = x|x| = \begin{cases} - |x^2| & \text{for } x < 0 \\ 0 & \text{for } x = 0 \\ |x^2| & \text{for } x > 0 \end{cases} = \text{sgn} (x) |x^2| \quad \Rightarrow f^{-1}(x) = \boxed{\text{sgn} (x)\sqrt{|x|}}

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