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Find a conjugation


(10 pts)

In this assignment, you’ll find a function \varphi that conjugates a given function g to another function f. Choose your name from the drop down menu below for the specifics:


For the functions f(x)=3x-3x^2 and g(x)=x^2+13x+35 , I found a conjugacy \varphi(x) =- \frac{1}{3}x - \frac{5}{3}

Directly evaluating:
f \circ \varphi = 3( - \frac{1}{3}x - \frac{5}{3})-3(- \frac{1}{3}x - \frac{5}{3})^2
f \circ \varphi = - \frac{x^2}{3}- \frac{13}{3}x-\frac{40}{3}
\varphi \circ g = -\frac{x^2}{3}+13 (\frac{-x}{3})+35(\frac{-1}{3})-\frac{40}{3}
\varphi \circ g = - \frac{x^2}{3}- \frac{13}{3}x-\frac{40}{3}
Thus, f \circ \varphi = \varphi \circ g.

The function’s cobweb plots behave similarly when beginning from their respective critical points, indicating the functions may be related to one another through a conjugation function.

Note that the functions are inverted and scaled differently.


For the functions f(x)=4x-4x^2 and g(x)=x^2+8x+10, I found a conjugacy \varphi(x)=-\frac{1}{4}x-\frac{1}{2}.

Directly evaluating:
\varphi\circ g=-\frac{1}{4}(x^2+8x+10)-\frac{1}{2}
\varphi\circ g=-\frac{1}{4}x^2-2x-3
Thus f\circ\varphi=\varphi\circ g.

The cobweb plots of the two functions behave similarly when they start orbiting from their critical points. The two function’s are inverted, scaled, and translated.


a. Let

f(x)=4x-4x^2 \space\text{and}\space g(x)=-\frac{4}{3}x^2-\frac{20}{3}x-\frac{28}{3}.

Consider the conjugacy \varphi(x)=3x-4. Evaluating \varphi\space\circ\space f we see,

\varphi\space\circ\space f=\varphi(f(x))\\ =3(4x-4x^2)-4\\ =-12x^2+12x-4.

Now we check g\space\circ\space\varphi,

g\space\circ\space\varphi=g(\varphi(x))\\ =-\frac{4}{3}(3x-4)^2-\frac{20}{3}(3x-4)-\frac{28}{3}\\ =-\frac{4}{3}(9x^2-24x+16)-\frac{20}{3}(3x-4)-\frac{28}{3}\\ =-12x^2+32x-\frac{64}{3}-20x+\frac{80}{3}-\frac{28}{3}\\ =-12x^2+12x-4.

Thus, since g\space\circ\space\varphi =\varphi\space\circ\space f, \varphi is a conjugacy from g to f.


My functions: f(x)=3x-3x^2 and g(x)=\frac{3x^2}{4} +\frac{21x}{2} +\frac{115}{4}
I found a conjugacy \varphi(x)=-\frac{1}{4}x -\frac{5}{4}.
I decided to directly evaluate f \circ \varphi and \varphi \circ g instead of g \circ \varphi and \varphi \circ f because it made my computations a little easier and it still worked since I kept my order consistent.
Directly evaluating:
f \circ \varphi=3(-\frac{1}{4}x - \frac{5}{4}) -3(-\frac{1}{4}x - \frac{5}{4})^2
f \circ \varphi=-\frac{3}{16}x^2 -\frac{21}{8}x -\frac{135}{16}
\varphi \circ g=-\frac{1}{4}(\frac{3x^2}{4} +\frac{21x}{2} +\frac{115}{4})-\frac{5}{4}
\varphi \circ g-\frac{3}{16}x^2 -\frac{21}{8}x -\frac{135}{16}
Thus, f \circ \varphi=\varphi \circ g.
Using 2 side by side cobweb plots it is easy to see the geometric similarity of the critical orbits of f and g.

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a. Given f(x) = x^2 - 1 and g(x) = \frac {x^2}{4}+\frac{5x}{2}-\frac{11}{4}, consider the conjugate function \varphi(x)=4x-5.
Directly evaluating \varphi \circ f, we see:
\varphi \circ f =4(x^2-1)-5=4x^2-4-5=4x^2-9.
Evaluating g\circ\varphi = \frac {(4x-5)^2}{4}+\frac{5(4x-5)}{2}-\frac{11}{4} =4x^2-10x+\frac{25}{4}+10x-\frac{25}{2}-\frac{11}{4} =4x^2-9, which shows \varphi\circ f=g\circ\varphi.
b. Beginning at their critical points, the cobweb plots for these functions both show the same activity, which in this case is a period 2 orbit.