Which of the following function is surjective but not injective.
(a) `f:R->R ,f(x)=x^4+2x^3-x^2+1`
(b) `f:R->R ,f(x)=x^2+x+1`
(c) `f:R->R^+ ,f(x)=sqrt(x^2+1)`
(d) `f:R->R ,f(x)=x^3+2x^2-x+1`

Options

A. `f:RrarrR, f(x)=x^(4)+2x^(3)-x^(2)+1`
B. `f:RrarrR,f(x)=x^(3)+x+1`
C. `f:RrarrR,f(x)=sqrt(1+x^(2))`
D. `f:RrarrR, f(x)=x^(3)+2x^(2)-x+1`

Answer

D

Explanation

`f(x)=x^(4)+2x^(3)-x^(2)+1` is polynomial of even degree hence its range can't be R.
hence not surjective
`f(x)=x^(3) =x^(3)+x+1,f'(x)=3x^(2)+1`
implies monotonic hence bijective.
`f(x) -sqrt(1+x^(2)),`
neither surjective not injective.
`f(x)=x^(3)+2x^(2)-x+1`
`f'(x)=3x^(2)+4x-1 implies D=16+12gt0`.
Hence f(x) is non-monotonic cubic polynomial hence surjective but not injective