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MATH / Let f(z)=(z+a)/(z+b) and g(z)=f(f(z)), where a and b are complex numbers. Suppose that |a|=1 and g(g(z))=z for all z for which g(g(z)) is defined. What is the difference between th…

Problem

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problem

Let f(z)=z+az+bf(z)= \frac{z+a}{z+b} and g(z)=f(f(z))g(z)=f(f(z)), where aa and bb are complex numbers. Suppose that |a|=1\left| a \right| = 1 and g(g(z))=zg(g(z))=z for all zz for which g(g(z))g(g(z)) is defined. What is the difference between the largest and smallest possible values of |b|\left| b \right|?
Plain-text mathematical notation (without MathML)
Let f(z)=(z+a)/(z+b) and g(z)=f(f(z)), where a and b are complex numbers. Suppose that |a|=1 and g(g(z))=z for all z for which g(g(z)) is defined. What is the difference between the largest and smallest possible values of |b|?
Original LaTeX notation
Let $f(z)= \frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $\left| a \right| = 1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $\left| b \right|$?

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
9d390453-e54b-5103-88cb-bc3dad07514f
task_key
intermediate~5falgebra--test--9d390453-e54b-5103-88cb-bc3dad07514f
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import