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Question

For real x,iff(x)=x3+5x+1, then


A

f is one-one but not onto R

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B

f is onto R but not one-one

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C

fis one-one and onto R

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D

fis neither one-one nor onto R

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Solution

The correct option is C

fis one-one and onto R


The explanation for the correct option:

Step1. Find whether the function is one-one or not:

f(x)=x3+5x+1f'(x)=3x2+5d(xn)dx=nxn-1f'(x)>0[3x2+5>0,xR]

xR,f(x) is strictly increasing function.

xR,f(x) is a one-one function,

Step2. Find whether the function is onto or not:

f(x) is polynomial, continuous, and increasing function.

yxRf(x)=y[f(x)isapolynomialofodddegreey(codomain)xR]limx+f(x)=limx+x3+5x+1=+limx-f(x)=limx-x3+5x+1=-

Rangeoff(x)=(-,+)=R=Codomain

f(x) is onto function.

Hence, Option(C) is the correct answer.


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