For real x, let `f(x)""=""x^3+""5x""+""1` , then (1) f is oneone but not onto R (2) f is onto R but not oneone (3) f is oneone and onto R (4) f is neither oneone nor onto R

Options

A. `f` is one-one but not onto R
B. `f` is onto R but not one-one
C. `f` is one-one and onto R
D. `f` is neither one-one nor onto R

Answer

C

Explanation

Given `f(x)=x^(3)+5x+1`
Now `f'(x) =3x^(2)+5 gt 0 AA x in R`
Therefore, `f(x)` is a strictly increasing function, and so it is one-one.
Clearly, `f(x)` is a continuous function and also increasing on R.
`underset (x to -oo)(lim)f(x)= -oo and underset(xto oo)(lim)f(x)=oo`
Hence, `f(x)` takes every volue between `-oo and oo .`
Thus, `f(x)` is an onto function.