If $f$ is a continuous function $f: mathbbR to mathbbR$ with property that $f(f(a)) = a$ for some $a in mathbbR$, $f(x) = x$ has …

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My Question:
Suppose $f$ is a continuous function $f:mathbbRtomathbbR$ and $f(f(a)) = a$ for some $ainmathbbR$ then the equation $f(x) = x$ has:



A) no real roots



B) exactly one real root



C) at most one real root



D) 3 real roots (at least)



Answer of the above is option D)



My process:



I have got infinite real roots if I keep $f(x)=x$ as function. Now I am stuck after this but deduced that all the even functions will be the answer as codomain is real numbers.




algebra-precalculus




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  • Thanks for editing it well.
    – jayant98
    Aug 13 at 8:50






  • 2




    Without additional hypothesis, option D) is not the right answer in general. Take the example of $f(x)=2x$.
    – mathcounterexamples.net
    Aug 13 at 8:51










  • An example with only two roots is the function $f(x)=e^x-1$ (take $a=0$)
    – Andrea Mori
    Aug 13 at 8:54










  • Yes, I think you are right. But it was given in my half yearly exam and teacher said that option was the right answer. I don't know how it will be answer.
    – jayant98
    Aug 13 at 8:54






  • 1




    @jayant98: that the image of $f$ must be $Bbb R$ is written nowhere in your question.
    – Andrea Mori
    Aug 13 at 9:24














up vote
0
down vote

favorite












My Question:
Suppose $f$ is a continuous function $f:mathbbRtomathbbR$ and $f(f(a)) = a$ for some $ainmathbbR$ then the equation $f(x) = x$ has:



A) no real roots



B) exactly one real root



C) at most one real root



D) 3 real roots (at least)



Answer of the above is option D)



My process:



I have got infinite real roots if I keep $f(x)=x$ as function. Now I am stuck after this but deduced that all the even functions will be the answer as codomain is real numbers.




algebra-precalculus




share|cite|improve this question






















  • Thanks for editing it well.
    – jayant98
    Aug 13 at 8:50






  • 2




    Without additional hypothesis, option D) is not the right answer in general. Take the example of $f(x)=2x$.
    – mathcounterexamples.net
    Aug 13 at 8:51










  • An example with only two roots is the function $f(x)=e^x-1$ (take $a=0$)
    – Andrea Mori
    Aug 13 at 8:54










  • Yes, I think you are right. But it was given in my half yearly exam and teacher said that option was the right answer. I don't know how it will be answer.
    – jayant98
    Aug 13 at 8:54






  • 1




    @jayant98: that the image of $f$ must be $Bbb R$ is written nowhere in your question.
    – Andrea Mori
    Aug 13 at 9:24












up vote
0
down vote

favorite









up vote
0
down vote

favorite











My Question:
Suppose $f$ is a continuous function $f:mathbbRtomathbbR$ and $f(f(a)) = a$ for some $ainmathbbR$ then the equation $f(x) = x$ has:



A) no real roots



B) exactly one real root



C) at most one real root



D) 3 real roots (at least)



Answer of the above is option D)



My process:



I have got infinite real roots if I keep $f(x)=x$ as function. Now I am stuck after this but deduced that all the even functions will be the answer as codomain is real numbers.




algebra-precalculus




share|cite|improve this question














My Question:
Suppose $f$ is a continuous function $f:mathbbRtomathbbR$ and $f(f(a)) = a$ for some $ainmathbbR$ then the equation $f(x) = x$ has:



A) no real roots



B) exactly one real root



C) at most one real root



D) 3 real roots (at least)



Answer of the above is option D)



My process:



I have got infinite real roots if I keep $f(x)=x$ as function. Now I am stuck after this but deduced that all the even functions will be the answer as codomain is real numbers.





algebra-precalculus





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 13 at 8:43









N. F. Taussig

38.5k93053




38.5k93053










asked Aug 13 at 8:36









jayant98

288




288











  • Thanks for editing it well.
    – jayant98
    Aug 13 at 8:50






  • 2




    Without additional hypothesis, option D) is not the right answer in general. Take the example of $f(x)=2x$.
    – mathcounterexamples.net
    Aug 13 at 8:51










  • An example with only two roots is the function $f(x)=e^x-1$ (take $a=0$)
    – Andrea Mori
    Aug 13 at 8:54










  • Yes, I think you are right. But it was given in my half yearly exam and teacher said that option was the right answer. I don't know how it will be answer.
    – jayant98
    Aug 13 at 8:54






  • 1




    @jayant98: that the image of $f$ must be $Bbb R$ is written nowhere in your question.
    – Andrea Mori
    Aug 13 at 9:24
















  • Thanks for editing it well.
    – jayant98
    Aug 13 at 8:50






  • 2




    Without additional hypothesis, option D) is not the right answer in general. Take the example of $f(x)=2x$.
    – mathcounterexamples.net
    Aug 13 at 8:51










  • An example with only two roots is the function $f(x)=e^x-1$ (take $a=0$)
    – Andrea Mori
    Aug 13 at 8:54










  • Yes, I think you are right. But it was given in my half yearly exam and teacher said that option was the right answer. I don't know how it will be answer.
    – jayant98
    Aug 13 at 8:54






  • 1




    @jayant98: that the image of $f$ must be $Bbb R$ is written nowhere in your question.
    – Andrea Mori
    Aug 13 at 9:24















Thanks for editing it well.
– jayant98
Aug 13 at 8:50




Thanks for editing it well.
– jayant98
Aug 13 at 8:50




2




2




Without additional hypothesis, option D) is not the right answer in general. Take the example of $f(x)=2x$.
– mathcounterexamples.net
Aug 13 at 8:51




Without additional hypothesis, option D) is not the right answer in general. Take the example of $f(x)=2x$.
– mathcounterexamples.net
Aug 13 at 8:51












An example with only two roots is the function $f(x)=e^x-1$ (take $a=0$)
– Andrea Mori
Aug 13 at 8:54




An example with only two roots is the function $f(x)=e^x-1$ (take $a=0$)
– Andrea Mori
Aug 13 at 8:54












Yes, I think you are right. But it was given in my half yearly exam and teacher said that option was the right answer. I don't know how it will be answer.
– jayant98
Aug 13 at 8:54




Yes, I think you are right. But it was given in my half yearly exam and teacher said that option was the right answer. I don't know how it will be answer.
– jayant98
Aug 13 at 8:54




1




1




@jayant98: that the image of $f$ must be $Bbb R$ is written nowhere in your question.
– Andrea Mori
Aug 13 at 9:24




@jayant98: that the image of $f$ must be $Bbb R$ is written nowhere in your question.
– Andrea Mori
Aug 13 at 9:24















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