I'm taking Abstract Algebra, and we're currently covering isometries of the Real and Complex plane. I'm going through and studying for our first midterm, and I'm working on a problem that asks to show that the composition of two functions composed with their inverses is a translation. The question gives the two functions $f(z)$ and $g(z)$, where $f(z) = (1+i)z + (1-i)$. How does one find the inverse of a function with complex variables? Or am I just overthinking it and the answer is not as hard as I think it is?
2026-03-28 10:35:48.1774694148
Inverse of a Function with Complex Variables
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As Sangchul Lee says, you find the inverse of a function of a complex variable exactly like you do a function of a complex variable. What I would do, since I prefer to have both "function" and "inverse function" as function of z, is write the equation as w= (1+i)z+ (1-i) and then swap w and z: z= (1+ i)w+ (1-i) and solve for w.