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g(x) = [(x+5)^2]/36

I got to the point where i reconized that the inverse of g(x) = 6sqrt(x) - 5 which looks alot like the function f(x), any hints or solutions to this problem?

- Thread starter SeReNiTy
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- #1

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g(x) = [(x+5)^2]/36

I got to the point where i reconized that the inverse of g(x) = 6sqrt(x) - 5 which looks alot like the function f(x), any hints or solutions to this problem?

- #2

HallsofIvy

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The only way I see of finding the points of intersection is to set f(x)= g(x), square both sides to get rid of the square root, and solve the resulting fourth degree equation

- #3

TD

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I don't know if you can get there your way (something with the inverse) but tehcnically, it's solvable since you get a 4th-degree polynomial. It won't be 'fun' though. Of course, watch out for introducing solution when squaring.

[tex]\frac{{\left( {x + 5} \right)^2 }}

{{36}} = 6\sqrt x \Leftrightarrow \left( {\frac{{\left( {x + 5} \right)^2 }}

{{36}}} \right)^2 = \left( {6\sqrt x } \right)^2 \Leftrightarrow \frac{{\left( {x + 5} \right)^4 }}

{{1296}} - 36x = 0[/tex]

If you'd want to know, mathematica gives me:

[tex]\begin{gathered}-5 + \frac{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}{2} - \hfill \\

\frac{{\sqrt{\frac{-{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}} -

216\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}} +

\frac{279936}

{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}}

{3}}}}{2}\end{gathered} [/tex]

and

[tex]\begin{gathered}-5 + \frac{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}{2} + \hfill \\

\frac{{\sqrt{\frac{-{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}} -

216\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}} +

\frac{279936}

{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}}

{3}}}}{2}\end{gathered} [/tex]

which are approx: [itex]x \to 0.013452\, \wedge \,x \to 29.150[/itex]

[tex]\frac{{\left( {x + 5} \right)^2 }}

{{36}} = 6\sqrt x \Leftrightarrow \left( {\frac{{\left( {x + 5} \right)^2 }}

{{36}}} \right)^2 = \left( {6\sqrt x } \right)^2 \Leftrightarrow \frac{{\left( {x + 5} \right)^4 }}

{{1296}} - 36x = 0[/tex]

If you'd want to know, mathematica gives me:

[tex]\begin{gathered}-5 + \frac{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}{2} - \hfill \\

\frac{{\sqrt{\frac{-{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}} -

216\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}} +

\frac{279936}

{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}}

{3}}}}{2}\end{gathered} [/tex]

and

[tex]\begin{gathered}-5 + \frac{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}{2} + \hfill \\

\frac{{\sqrt{\frac{-{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}} -

216\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}} +

\frac{279936}

{{\sqrt{\frac{{\left( 29386561536 - 120932352\,{\sqrt{57549}} \right) }^{\frac{1}{3}}}{3} +

72\,2^{\frac{2}{3}}\,{\left( 3\,\left( 243 + {\sqrt{57549}} \right) \right) }^{\frac{1}{3}}}}}}

{3}}}}{2}\end{gathered} [/tex]

which are approx: [itex]x \to 0.013452\, \wedge \,x \to 29.150[/itex]

Last edited:

- #4

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Oh, with the inverse thing, i just remembered doing a question a while ago that involved finding the points of intersection between two inverse functions, it was much easier to computer the intersection between one of the functions and y = x since inverse always intersect along that line.HallsofIvy said:

The only way I see of finding the points of intersection is to set f(x)= g(x), square both sides to get rid of the square root, and solve the resulting fourth degree equation

How do i go about solving the 4th degree polynomial?

- #5

TD

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Just as there are formula's for the quadratic and cubic, there also exists one for the 4th-degree, named Ferrari.SeReNiTy said:How do i go about solving the 4th degree polynomial?

http://mathworld.wolfram.com/QuarticEquation.html

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