If you couldn't already tell, one of my favorite things about math are the "unconventional" functions, like greatest common divisor, least common multiple, mean, and modulo. For simplicity, I'll refer to these functions as the "strange functions". I know they're not actually called that, but I don't really want to write all four "strange functions" over and over again.
Anyways, a while ago, I made a few graphs in Desmos using almost exclusively the "strange functions", basic arithmetic, and exponents. I used a few logarithms and absolute values, but I used logarithims in a total of 4 functions out of all 44, and absolute value in only 1 function. There was always at least one gcd or lcm in each function. Most of them were written in the style of "functions(x,y) = 1", but a few (especially in the third graph) were written in the style of "functions(x,y) = other_functions(x,y)", and one function in the second graph was written in the style of "|y| = functions(|x|, |1/x|)". Every single graph turned out... odd, to say the least. Could someone please provide an explanation as to why the graphs look this way? Here are the links to the graphs: Graph 1, Graph 2, and Graph 3. Here are pictures of the graphs:
Graph 1:
Graph 2:
Graph 3:
Here is each function individually, the graph of that function, and what is confusing about that graph:
First Graph Functions:
1. $gcd(x^1,y^1)=1$
First things first, why are there squares going away from the origin with a slope of either 1 or negative 1 in each quadrant? Second of all, why are all vertices in the graph at a non-integer multiple of ${1 \over 2}$?
2. $lcm(x^1,y^1)=1$
Why are there only lines at $\pm {3 \over 4}$?
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Second Graph Functions:
1. Function 1
2. Function 2.
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Third Graph Functions:
1. Function 1
2. Function 2.
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2026-02-24 07:04:51.1771916691