- #1

SeReNiTy

- 170

- 0

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 a lot like the function f(x), any hints or solutions to this problem?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter SeReNiTy
- Start date

- #1

SeReNiTy

- 170

- 0

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 a lot like the function f(x), any hints or solutions to this problem?

- #2

HallsofIvy

Science Advisor

Homework Helper

- 43,021

- 970

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

Homework Helper

- 1,022

- 0

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

SeReNiTy

- 170

- 0

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

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.

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

- #5

TD

Homework Helper

- 1,022

- 0

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

Share:

- Last Post

- Replies
- 1

- Views
- 300

- Replies
- 21

- Views
- 441

- Replies
- 2

- Views
- 324

- Replies
- 1

- Views
- 397

- Replies
- 7

- Views
- 305

- Replies
- 11

- Views
- 327

- Replies
- 2

- Views
- 249

- Replies
- 7

- Views
- 293

- Replies
- 11

- Views
- 362

- Replies
- 6

- Views
- 425