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If f(x) is defined on [0, 1] by the rul...

If `f(x)` is defined on `[0, 1]` by the rule `f(x)={x,` if `x` is rational ,`1-x,` if `x` is rational ' then for all `x in [0,1]` ,`f(f(x))` is

A

x

B

`-x`

C

`1+x`

D

1-x

Text Solution

Verified by Experts

The correct Answer is:
A
We have, `f(x)={{:(x","if x in Q ),(1-x"," if x in Q):}`
`f(f(x))={{:(f(x)"," x in Q ),(f(1-x)"," x in Q):}`
`f(f(x))={{:(x"," x in Q ),(1-(1-x)"," x in Q):} " " [{:( :' x in Q implies 1-x in Q),( :' f(1-x)=1-(1-x)):}]`
`implies f(f(x))={{:(x"x"x in Q),(x","x in Q):}`
`implies f(f(x))=x` for all ` x in [0,1]`
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OBJECTIVE RD SHARMA ENGLISH-REAL FUNCTIONS -Chapter Test
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  2. The period of the function f(x)=sin^(4)3x+cos^(4)3x, is

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  3. The value of integer n for which the function f(x)=(sinx)/(sin(x / n)...

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  4. The period of the function f(x)=sin((2x+3)/(6pi)), is

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  5. The domain of the function f(x)=sqrt(log((1)/(|sinx|)))

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  6. The domain of the function f(x)=log(10) (sqrt(x-4)+sqrt(6-x)) is :

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  7. Let f(x)=(sqrt(sinx))/(1+(sinx)^(1/3)) then domain f contains

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  8. If f : R -> R is defined by f(x) = [2x] - 2[x] for x in R, where [x] i...

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  9. If N denotes the set of all positive integers and if f : N -> N is def...

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  10. The set of value of a for which the function f(x)=sinx+[(x^(2))/(a)] d...

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  11. If f(x)={{:(-1, x lt 0),(0, x=0 and g(x)=x(1-x^(2))", then"),(1, x gt ...

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  12. Find the equivalent definition of f(x)=max.{x^(2),(1-x)^(2),2x(1-x)...

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  13. If f(x) is defined on [0,1], then the domain of f(3x^(2)) , is

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  14. The function f(x) is defined in [0,1] . Find the domain of f(t a nx)do...

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  15. The domain of definition of the real function f(x)=sqrt(log(12)x^(2)) ...

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  16. The values of ba n dc for which the identity of f(x+1)-f(x)=8x+3 is sa...

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  17. The function f(x)=sin""(pix)/(2)+2 cos ""(pix)/(3)-tan""(pix)/(4) is p...

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  18. The period of the function sin""((pix)/(2))+cos((pix)/(2)), is

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  19. If x in R, then f(x)=sin^(-1)((2x)/(1+x^(2))) is equal to

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  20. If x in R , then f(x)=cos^(-1)((1-x^(2))/(1+x^(2))) is equal to

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  21. The equivalent definition of the function f(x)=lim(n to oo)(x^(n)-x^(-...

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