If the following functions are defined from `[-1,1]to[-1,1],` select those which are not bijective.

Options

A. `sin(sin^(-1)x)`
B. `(2)/(pi) sin^(-1)(sin x)`
C. `(sgn(x))ln (e^(x))`
D. `x^(3)(sgn(x))`

Answer

B, C, D

Explanation

`f(x)=sin(sin^(-1)x)=x AA x in [-1,1],` which is one-one and onto.
`f(x)=(2)/(pi)sin^(-1)(sinx)=(2)/(pi)x`
The range of the function for `x in [-1,1] " is " [-(2)/(pi),(2)/(pi)],`
which is a subset of [-1, 1].
Hence, the function is one-one but not onto and hence, is not bijective.
`f(x)=(sgn(x))"In"(e^(x))=(sgn(x))x={(x",",x gt 0),(-x",",x lt 0),(0",",x=0):}.`
This function has range [0, 1] which is a subset of [-1, 1].
Hence, the function is into. Also, the function is may-one.
`f(x)=x^(3)sgn(x)={(x^(3)",", x gt 0),(-x^(3)",", x lt 0),(0",",x=0):}`
which is many-one and into.