UPSEE Solved Model Paper 7

Given
f (x) =

x|x|

1+x2

We have to find the inverse function f−1 (x)
Let
y = f (x) = {

x2 1+x2	x≥0
−x2 1+x2	x<0

If y =

x2

1+x2

⇒ x2−x2y−y = 0
⇒ x2 =

y

1−y

⇒ x = √

y

1−y

for 0 ≤ y < 1
Similarly, if y =

−x2

1+x2

⇒ x = −√

−y

1+y

for - 1 < y < 0
Since f−1(y) = {

−√ −y 1+y	−1<y<0
√ y 1−y	0≤y<1

Replacing y by x, we get
f−1(x) = {

−√ −x 1+x	−1<x<0
√ x 1−x	0≤x<1

Hence, option 'D' is correct.