Trying desparately to wrap my head around proportional relationships

[Ontophile]

Homework Statement

A) Suppose that we have 3 quantities, x, y, and z. An increase in x by a factor of F is followed by a subsequent increase of y by a factor of 2F. Furthermore, the very same increase in x by a factor of F results in a subsequent decrease of z by a factor of F/2. Similarly, when x is decreased by a factor of F, y is decreased by a factor of F/2 and z is increased by a factor of 2F. How do I mathematically formalize this relationship between x, y, and z?

B) Now suppose that an increase in x by a factor of F yields an increase in y by a factor of F² and a decrease in z by a factor of √F (I assume that that's a square root symbol). How do I mathematically formalize THIS relationship between x, y, and z?

C) Lastly, suppose that an increase in x by a factor of F increases y by a factor of F and decreases z by a factor of F? How do I mathematically formalize THIS relationship between x, y, and z?

 

[Simon Bridge]

Note, if y were proportional to x, then increasing x by a factor of F would also increase y by a factor of F^q

if y = x² then increasing x by F increases y by F²
If q=1 then y is directly proportional to x
If q < 0 they y is inversely proportional to x
The proportionality is characterized by q as in "inverse-square" for q=-2.

the described, increase by F leads to increase by 2F means
if we start at x=1, and put y(1)=a
then at x=2, x has doubled for F=2, so y must have quadrupled: 2F=4 to 4a
at x = 3, F=3, so 2F=6, so y(3)=6a;
continuing gives:

x = {1, 2, 3, 4, 5,...}
y = {a, 4a, 6a, 8a, 10a,...}

x(4) = 4x(1) means y(4)=8y(1): which is what we have above
x(4) = 2x(2) means y(4)=4y(2) means 4.4a=8a - which is false.

since 8a ≠ 16a, this interpretation of "increase by a factor" is flawed.

If it just means x→x+F => y -> y→2F, then this is satisfied by y=2x
But we are also told that x→x-F => y→y-F/2 (by the same interpretation).
This won't work for y=2x.

I can see how you got confused! What exactly does "increase/decrease by a factor" mean in this situation?

Can we find q so that F^q = 2F
well: this means F^(q-1)=2
which means q-1 = log_F(2) = \frac{\ln(2)}{\ln(F)}
suggesting that the value of q depends on the value of F.

eg, if F=2 the q=2, if F=4, q=3/2
so the answer is "no" and we are back with:
What exactly does "increase/decrease by a factor" mean in this situation?

We can see this when we realize we are looking for y=f(x) with the property that f(Fx)=2F.f(x)
When F = 1 (no change) this reduces to f(x)=2f(x) which is false.

ref:
----------------------------
http://www.phy.syr.edu/courses/PHY106/Slides/PPT/Lec2-Proportionality-Algebra.pdf