Find the inverse of the function `f: [-1,1] to [-1,1],f(x) =x^(2) xx sgn (x).`

Explanation

`f(x) =x^(2) xx sgn (x)`
`={(x^(2)(1)",", x gt 0),(0"," , x=0),(x^(2)(-1)",", x lt 0):}`
` :. f(x) ={(x^(2)",", 0le x le1),(-x^(2)",", -1 le x lt0):}`
Now `y=x^(2),0 le x le 1`
Since `y in [0,1],` we have `x=sqrt(y)`
For `y=-x^(2),-1 le x le 0.`
Since `y in [-1,0], " we have " x= -sqrt(-y)`
Thus, `f^(-1)(x)={(sqrt(x)",", 0le x le1),(-sqrt(-x)",", -1 le x lt0):}`