Find equation w/ 3 Fixed points

May 2010
10
0
baltimore, md
Hello

I am returning to school after a long break. One of my assignments is a mix of ODE and discrete math and is clogging my mind.

I have been struggling on my 12 question homework for 4 hours now and have made it 1/4 of the way through. Luckily i have found this forum and hope someone(s) can help.

The question is this:
Find a continuous function:
f : [0, 1] --> [0, 1] with exactly three fixed points and draw its graph.

The fixed points is throwing me for a loop.(Headbang)

Any suggestions??
 

undefined

MHF Hall of Honor
Mar 2010
2,340
821
Chicago
Hello

I am returning to school after a long break. One of my assignments is a mix of ODE and discrete math and is clogging my mind.

I have been struggling on my 12 question homework for 4 hours now and have made it 1/4 of the way through. Luckily i have found this forum and hope someone(s) can help.

The question is this:
Find a continuous function:
f : [0, 1] --> [0, 1] with exactly three fixed points and draw its graph.

The fixed points is throwing me for a loop.(Headbang)

Any suggestions??
Start with definition: x is a fixed point of a function f if and only if f(x) = x.

All you have to do is draw a continuous function in the region \(\displaystyle x \in [0,1], y \in [0,1]\) that intersects with the line y=x three times. If you start at the point (0,0) and end at the point (1,1) then you can just have it cross the line y=x once for \(\displaystyle x \in (0,1)\).

You can make it piecewise...
 
Nov 2009
485
184
How about \(\displaystyle f(x)=x^3\)? We have \(\displaystyle f(-1)=-1\), \(\displaystyle f(0)=0\), \(\displaystyle f(1)=1\). These are all of the fixed points since \(\displaystyle x^3=x\) has only three solutions, by the Fundamental Theorem of Algebra.
 

undefined

MHF Hall of Honor
Mar 2010
2,340
821
Chicago
How about \(\displaystyle f(x)=x^3\)? We have \(\displaystyle f(-1)=-1\), \(\displaystyle f(0)=0\), \(\displaystyle f(1)=1\). These are all of the fixed points since \(\displaystyle x^3=x\) has only three solutions, by the Fundamental Theorem of Algebra.
But when restricted to the domain [0,1], f(x)=x^3 only has two fixed points.
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
But when restricted to the domain [0,1], f(x)=x^3 only has two fixed points.
Scale, \(\displaystyle f\left(\frac{x+1}{2}\right)\)
 

undefined

MHF Hall of Honor
Mar 2010
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Scale, \(\displaystyle f\left(\frac{x+1}{2}\right)\)
Maybe I'm misinterpreting.. \(\displaystyle f(x) = x^3\) so \(\displaystyle f\left(\frac{x+1}{2}\right)=\left(\frac{x+1}{2}\right)^3\) which has only two fixed points for \(\displaystyle x \in [0,1]\) given by \(\displaystyle x \in \{-2+\sqrt{5},1\}\).

 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
Maybe I'm misinterpreting.. \(\displaystyle f(x) = x^3\) so \(\displaystyle f\left(\frac{x+1}{2}\right)=\left(\frac{x+1}{2}\right)^3\) which has only two fixed points for \(\displaystyle x \in [0,1]\) given by \(\displaystyle x \in \{-2+\sqrt{5},1\}\).

Oh! fixed points! ....why don't you just construct a polygonal line that intersects the line \(\displaystyle y=x\) three times?
 
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May 2010
10
0
baltimore, md
Oh! fixed points! ....why don't you just construct a polygonal line that intersects the line \(\displaystyle y=x\) three times?
that would be a piecewise function correct?

i was just hoping for something a little more "pretty" than that
 

undefined

MHF Hall of Honor
Mar 2010
2,340
821
Chicago
that would be a piecewise function correct?

i was just hoping for something a little more "pretty" than that
My original thought, before the piecewise idea, was to transform the sine curve so that it's rotated 45 degrees and can be stretched or compressed (along the line y=x) to produce as many fixed points as desired (greater than or equal to 2)... but since I figured you might ask how this is done, and I'd have to think about that, I just went the easier route. :)