Systems of Linear Functions and their Reciprocals

Recently, I’ve been exploring the relationship between systems of linear functions and their reciprocals, namely:

20130715 Function Set Reciprocal

The following are the two methodologies I used in this investigation:

1)    Lo-tech method (longer way): Solve for all axis-intercepts, asymptotes and points of intersection for various examples.
2)    Higher-tech method (shorter way): Graph examples using software.

After sketching several examples, I noticed that at the points of intersection (poi) y = ±1.  This generalization was useful when calculating/checking the x-coordinate of the poi without the aid of graphing software.

20130715 Working 220130715 Working 1

After sketching the long way, similar results were confirmed through software.

20130715 x+5 and reciprocal

Calculating the poi algebraically:20130715 Function Set Reciprocal Poi 1

confirmed the observed coordinates for the poi:

20130715 Function Set Reciprocal Poi Coord 1

I wondered if this relationship could be extended to a system of functions such as:

20130715 Function Set Reciprocal 2

Plotting examples using graphing software didn’t yield any transparent relationship between the coefficients of the function and the poi coordinates.

20130715 x+5 and 7:x+5

So, again, calculating the poi algebraically:

20130715 Function Set Reciprocal Poi 2

resulted in the coordinates for the poi (with invariant point y = c^(1/2):

20130715 Function Set Reciprocal Poi Coord 2

Same same but different!

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