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Question

If the function: RR, is defined by fx=xx-sinx, then which of the following statements is TRUE?


A

Function is one-one, but not onto

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B

Function is onto, but not one-one

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C

Function is both one-one and onto

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D

Function is neither one-one nor onto

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Solution

The correct option is C

Function is both one-one and onto


Explanation for the correct option

Step 1: Identify the nature of the given function

Given function, fx=xx-sinx

Identifying if the function is odd, by applying the condition f(-x)=-f(x)

f(-x)=-xx-sinx-f(x)=-xx-sinx

So, f(-x)=-f(x) for the given function. This also demonstrates that the function is continuous.

Thus, fx=xx-sinx is an odd, non-periodic continuous function.

Step 2: Evaluate the domain and range of the function

The function can be written as f(x)=x2-xsinx,x0-x2+xsinx,x<0x=+x;x0-x;x<0

So,

f(x+)=limxx21-sinxxf(x+)=

and,

f(x-)=limx-x21-sinxxf(x-)=-

Thus, the range of f(x)=R and f(x) is an onto function.

Differentiate the given function, f(x)=x2-xsinx,x0-x2+xsinx,x<0, we get,

f'(x)=2x-sinx-xcosx,x0-2x+sinx+xcosx,x<0

f'x=x-sinx+(x1-cosx), where the result will always be positive or zero.

So, f'(x)>0x-,

Thus, f(x) is a one-one function.

Therefore, option (C) i.e. Function is both one-one and onto, is the correct answer.


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