Finding a function from the given equation











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Let f be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x in R^+$. Then which of the following is correct?

A) f is an even function

B) f is an odd function

C) $lim_{xto infty} frac{f(x)}{x^3}=1$

D $lim_{xto infty} bigg(frac{f(x)}{x^2}-x bigg)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.










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  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    59 mins ago






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    55 mins ago

















up vote
1
down vote

favorite
1













Let f be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x in R^+$. Then which of the following is correct?

A) f is an even function

B) f is an odd function

C) $lim_{xto infty} frac{f(x)}{x^3}=1$

D $lim_{xto infty} bigg(frac{f(x)}{x^2}-x bigg)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.










share|cite|improve this question







New contributor




Tony is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    59 mins ago






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    55 mins ago















up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1






Let f be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x in R^+$. Then which of the following is correct?

A) f is an even function

B) f is an odd function

C) $lim_{xto infty} frac{f(x)}{x^3}=1$

D $lim_{xto infty} bigg(frac{f(x)}{x^2}-x bigg)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.










share|cite|improve this question







New contributor




Tony is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












Let f be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x in R^+$. Then which of the following is correct?

A) f is an even function

B) f is an odd function

C) $lim_{xto infty} frac{f(x)}{x^3}=1$

D $lim_{xto infty} bigg(frac{f(x)}{x^2}-x bigg)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.







calculus limits functions even-and-odd-functions






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share|cite|improve this question







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Check out our Code of Conduct.






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Check out our Code of Conduct.












  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    59 mins ago






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    55 mins ago




















  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    59 mins ago






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    55 mins ago


















To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
– Kavi Rama Murthy
59 mins ago




To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
– Kavi Rama Murthy
59 mins ago




1




1




Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
– Sauhard Sharma
55 mins ago






Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
– Sauhard Sharma
55 mins ago












2 Answers
2






active

oldest

votes

















up vote
5
down vote



accepted










Hints: look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






share|cite|improve this answer























  • But the answer given is C).
    – Tony
    44 mins ago










  • I double checked my solution and it looks right. I think the answer given is wrong.
    – Kavi Rama Murthy
    41 mins ago










  • See Sauhard Sharma's comment. Can that be an explanation?
    – Tony
    32 mins ago












  • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
    – jayant98
    30 mins ago


















up vote
0
down vote













To expand on @KaviRamaMurthy's answer:
Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
$$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
$$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
$$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
$$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
$$f(x)=x^3+b_1x.$$



The only way (B) can hold is if they meant




for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







share|cite|improve this answer





















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    2 Answers
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    2 Answers
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    up vote
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    down vote



    accepted










    Hints: look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






    share|cite|improve this answer























    • But the answer given is C).
      – Tony
      44 mins ago










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      41 mins ago










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      32 mins ago












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      30 mins ago















    up vote
    5
    down vote



    accepted










    Hints: look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






    share|cite|improve this answer























    • But the answer given is C).
      – Tony
      44 mins ago










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      41 mins ago










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      32 mins ago












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      30 mins ago













    up vote
    5
    down vote



    accepted







    up vote
    5
    down vote



    accepted






    Hints: look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






    share|cite|improve this answer














    Hints: look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 5 mins ago

























    answered 50 mins ago









    Kavi Rama Murthy

    48.2k31854




    48.2k31854












    • But the answer given is C).
      – Tony
      44 mins ago










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      41 mins ago










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      32 mins ago












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      30 mins ago


















    • But the answer given is C).
      – Tony
      44 mins ago










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      41 mins ago










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      32 mins ago












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      30 mins ago
















    But the answer given is C).
    – Tony
    44 mins ago




    But the answer given is C).
    – Tony
    44 mins ago












    I double checked my solution and it looks right. I think the answer given is wrong.
    – Kavi Rama Murthy
    41 mins ago




    I double checked my solution and it looks right. I think the answer given is wrong.
    – Kavi Rama Murthy
    41 mins ago












    See Sauhard Sharma's comment. Can that be an explanation?
    – Tony
    32 mins ago






    See Sauhard Sharma's comment. Can that be an explanation?
    – Tony
    32 mins ago














    @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
    – jayant98
    30 mins ago




    @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
    – jayant98
    30 mins ago










    up vote
    0
    down vote













    To expand on @KaviRamaMurthy's answer:
    Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
    $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
    $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
    So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
    $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
    $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
    which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
    $$f(x)=x^3+b_1x.$$



    The only way (B) can hold is if they meant




    for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







    share|cite|improve this answer

























      up vote
      0
      down vote













      To expand on @KaviRamaMurthy's answer:
      Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
      $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
      $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
      So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
      $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
      $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
      which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
      $$f(x)=x^3+b_1x.$$



      The only way (B) can hold is if they meant




      for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        To expand on @KaviRamaMurthy's answer:
        Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
        $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
        $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
        So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
        $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
        $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
        which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
        $$f(x)=x^3+b_1x.$$



        The only way (B) can hold is if they meant




        for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







        share|cite|improve this answer












        To expand on @KaviRamaMurthy's answer:
        Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
        $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
        $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
        So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
        $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
        $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
        which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
        $$f(x)=x^3+b_1x.$$



        The only way (B) can hold is if they meant




        for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.








        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 28 mins ago









        Bjørn Kjos-Hanssen

        2,066818




        2,066818






















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