If-f-R-1-1-is-defined-by-f-x-x-x-1-x-2-then-prove-that-f-1-x-sgn-x-x-1-x-

Question Number 13595 by Tinkutara last updated on 21/May/17

If f : R \to (- 1, 1) is defined by

f (x) = \frac{- x ∣ x ∣}{1 + x^{2}}, then prove that

f^{- 1} (x) = - sgn (x) \sqrt{\frac{∣ x ∣}{1 - ∣ x ∣}}

Answered by mrW1 last updated on 21/May/17

f (x) = y = \frac{- x ∣ x ∣}{1 + x^{2}}

i f y ⩾ 0 \Rightarrow x ⩽ 0 \Rightarrow∣ x ∣= - x . . (i)

i f y ⩽ 0 \Rightarrow x ⩾ 0 \Rightarrow∣ x ∣= x . . (i i)

(i) :

y = \frac{x^{2}}{1 + x^{2}}

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

x = - \sqrt{\frac{y}{1 - y}} (f o r y ⩾ 0, x ⩽ 0)

(i i) :

y = - \frac{x^{2}}{1 + x^{2}}

(1 + y) x^{2} = - y

x = \sqrt{\frac{- y}{1 + y}} (f o r y ⩽ 0, x ⩾ 0)

f o r b o t h (i) a n d (i i) :

x = - s g n (y) \sqrt{\frac{∣ y ∣}{1 - ∣ y ∣}}

\Rightarrow f^{- 1} (x) = - s g n (x) \sqrt{\frac{∣ x ∣}{1 - ∣ x ∣}}

Commented by Tinkutara last updated on 21/May/17

Thanks Sir!