Range of a rational function with radicals

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Find the range of the function



$$frac65sqrtx^2-10x+29 - 2$$



I tried using inverses, but the equation got super messy and I dont think its a good method for this problem.



$frac65sqrtx^2-10x+29 - 2 = y$



getting the inverse,



$frac65sqrty^2-10y+29 - 2 = x$



$frac4x^2+24x+3625x^2= y^2-10y +29$



Then it would be a quadratic function in y, but the discriminant becomes really big



$100- 116(frac4x^2+24x+3625x^2)$







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  • What do you mean by "I tried using inverses"? Could you show your work?
    – A. Pongrácz
    Aug 19 at 9:43










  • A graph will show you some things that will help. It isn't hard to see that it can't be negative and a little calculus will help you find the single maximum.
    – Bruce
    Aug 19 at 9:49










  • Completing the square even saves you the calculus
    – Bruce
    Aug 19 at 9:52










  • You can also write $ y^2-10y +29=left ( 1 over 5left ( 6over x+2right ) right )^2$
    – N74
    Aug 19 at 17:05














up vote
2
down vote

favorite
1












Find the range of the function



$$frac65sqrtx^2-10x+29 - 2$$



I tried using inverses, but the equation got super messy and I dont think its a good method for this problem.



$frac65sqrtx^2-10x+29 - 2 = y$



getting the inverse,



$frac65sqrty^2-10y+29 - 2 = x$



$frac4x^2+24x+3625x^2= y^2-10y +29$



Then it would be a quadratic function in y, but the discriminant becomes really big



$100- 116(frac4x^2+24x+3625x^2)$







share|cite|improve this question






















  • What do you mean by "I tried using inverses"? Could you show your work?
    – A. Pongrácz
    Aug 19 at 9:43










  • A graph will show you some things that will help. It isn't hard to see that it can't be negative and a little calculus will help you find the single maximum.
    – Bruce
    Aug 19 at 9:49










  • Completing the square even saves you the calculus
    – Bruce
    Aug 19 at 9:52










  • You can also write $ y^2-10y +29=left ( 1 over 5left ( 6over x+2right ) right )^2$
    – N74
    Aug 19 at 17:05












up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





Find the range of the function



$$frac65sqrtx^2-10x+29 - 2$$



I tried using inverses, but the equation got super messy and I dont think its a good method for this problem.



$frac65sqrtx^2-10x+29 - 2 = y$



getting the inverse,



$frac65sqrty^2-10y+29 - 2 = x$



$frac4x^2+24x+3625x^2= y^2-10y +29$



Then it would be a quadratic function in y, but the discriminant becomes really big



$100- 116(frac4x^2+24x+3625x^2)$







share|cite|improve this question














Find the range of the function



$$frac65sqrtx^2-10x+29 - 2$$



I tried using inverses, but the equation got super messy and I dont think its a good method for this problem.



$frac65sqrtx^2-10x+29 - 2 = y$



getting the inverse,



$frac65sqrty^2-10y+29 - 2 = x$



$frac4x^2+24x+3625x^2= y^2-10y +29$



Then it would be a quadratic function in y, but the discriminant becomes really big



$100- 116(frac4x^2+24x+3625x^2)$









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 19 at 9:50

























asked Aug 19 at 9:41









SuperMage1

707210




707210











  • What do you mean by "I tried using inverses"? Could you show your work?
    – A. Pongrácz
    Aug 19 at 9:43










  • A graph will show you some things that will help. It isn't hard to see that it can't be negative and a little calculus will help you find the single maximum.
    – Bruce
    Aug 19 at 9:49










  • Completing the square even saves you the calculus
    – Bruce
    Aug 19 at 9:52










  • You can also write $ y^2-10y +29=left ( 1 over 5left ( 6over x+2right ) right )^2$
    – N74
    Aug 19 at 17:05
















  • What do you mean by "I tried using inverses"? Could you show your work?
    – A. Pongrácz
    Aug 19 at 9:43










  • A graph will show you some things that will help. It isn't hard to see that it can't be negative and a little calculus will help you find the single maximum.
    – Bruce
    Aug 19 at 9:49










  • Completing the square even saves you the calculus
    – Bruce
    Aug 19 at 9:52










  • You can also write $ y^2-10y +29=left ( 1 over 5left ( 6over x+2right ) right )^2$
    – N74
    Aug 19 at 17:05















What do you mean by "I tried using inverses"? Could you show your work?
– A. Pongrácz
Aug 19 at 9:43




What do you mean by "I tried using inverses"? Could you show your work?
– A. Pongrácz
Aug 19 at 9:43












A graph will show you some things that will help. It isn't hard to see that it can't be negative and a little calculus will help you find the single maximum.
– Bruce
Aug 19 at 9:49




A graph will show you some things that will help. It isn't hard to see that it can't be negative and a little calculus will help you find the single maximum.
– Bruce
Aug 19 at 9:49












Completing the square even saves you the calculus
– Bruce
Aug 19 at 9:52




Completing the square even saves you the calculus
– Bruce
Aug 19 at 9:52












You can also write $ y^2-10y +29=left ( 1 over 5left ( 6over x+2right ) right )^2$
– N74
Aug 19 at 17:05




You can also write $ y^2-10y +29=left ( 1 over 5left ( 6over x+2right ) right )^2$
– N74
Aug 19 at 17:05










2 Answers
2






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$frac65sqrt(x^2-10x+29)-2=frac65sqrt((x-5)^2+4)-2$



so you can see that there will be a maximum at $x=5$ and as $x$ tends to either positive or negative infinity the graph goes to zero.



So just plug in $x=5$ to get $frac34$ and you have a range $]0,frac34]$






share|cite|improve this answer



























    up vote
    7
    down vote













    Hint



    If you rewrite $x^2-10x+29=(x-5)^2+4$, it's easier to see that:
    $$(x-5)^2+4 in [4,+infty)$$
    $$sqrt(x-5)^2+4 in [2,+infty)$$
    $$5sqrt(x-5)^2+4-2 in [8,+infty)$$
    Does that help?






    share|cite|improve this answer




















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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      2
      down vote



      accepted










      $frac65sqrt(x^2-10x+29)-2=frac65sqrt((x-5)^2+4)-2$



      so you can see that there will be a maximum at $x=5$ and as $x$ tends to either positive or negative infinity the graph goes to zero.



      So just plug in $x=5$ to get $frac34$ and you have a range $]0,frac34]$






      share|cite|improve this answer
























        up vote
        2
        down vote



        accepted










        $frac65sqrt(x^2-10x+29)-2=frac65sqrt((x-5)^2+4)-2$



        so you can see that there will be a maximum at $x=5$ and as $x$ tends to either positive or negative infinity the graph goes to zero.



        So just plug in $x=5$ to get $frac34$ and you have a range $]0,frac34]$






        share|cite|improve this answer






















          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          $frac65sqrt(x^2-10x+29)-2=frac65sqrt((x-5)^2+4)-2$



          so you can see that there will be a maximum at $x=5$ and as $x$ tends to either positive or negative infinity the graph goes to zero.



          So just plug in $x=5$ to get $frac34$ and you have a range $]0,frac34]$






          share|cite|improve this answer












          $frac65sqrt(x^2-10x+29)-2=frac65sqrt((x-5)^2+4)-2$



          so you can see that there will be a maximum at $x=5$ and as $x$ tends to either positive or negative infinity the graph goes to zero.



          So just plug in $x=5$ to get $frac34$ and you have a range $]0,frac34]$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 19 at 9:54









          Bruce

          574113




          574113




















              up vote
              7
              down vote













              Hint



              If you rewrite $x^2-10x+29=(x-5)^2+4$, it's easier to see that:
              $$(x-5)^2+4 in [4,+infty)$$
              $$sqrt(x-5)^2+4 in [2,+infty)$$
              $$5sqrt(x-5)^2+4-2 in [8,+infty)$$
              Does that help?






              share|cite|improve this answer
























                up vote
                7
                down vote













                Hint



                If you rewrite $x^2-10x+29=(x-5)^2+4$, it's easier to see that:
                $$(x-5)^2+4 in [4,+infty)$$
                $$sqrt(x-5)^2+4 in [2,+infty)$$
                $$5sqrt(x-5)^2+4-2 in [8,+infty)$$
                Does that help?






                share|cite|improve this answer






















                  up vote
                  7
                  down vote










                  up vote
                  7
                  down vote









                  Hint



                  If you rewrite $x^2-10x+29=(x-5)^2+4$, it's easier to see that:
                  $$(x-5)^2+4 in [4,+infty)$$
                  $$sqrt(x-5)^2+4 in [2,+infty)$$
                  $$5sqrt(x-5)^2+4-2 in [8,+infty)$$
                  Does that help?






                  share|cite|improve this answer












                  Hint



                  If you rewrite $x^2-10x+29=(x-5)^2+4$, it's easier to see that:
                  $$(x-5)^2+4 in [4,+infty)$$
                  $$sqrt(x-5)^2+4 in [2,+infty)$$
                  $$5sqrt(x-5)^2+4-2 in [8,+infty)$$
                  Does that help?







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Aug 19 at 9:53









                  StackTD

                  20.5k1544




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