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For real x, let f(x)""=""x^3+""5x""+""1 ...

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

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

Text Solution

Verified by Experts

The correct Answer is:
C
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.
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Knowledge Check

  • If f:RtoR be defined as f(x)=x^(4) . Choose the corrent answer : (a) f is one-one onto (b)f is many-one onto (c)f is one-one but not onto (d)f is neither one-one nor onto

    A
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    B
    f is many-one onto
    C
    f is one-one but not onto
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    f is neither one-one nor onto