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Suppose $T$ is a linear fractional transformation such that $T(0)=1, T(1)=i,$ and $T(\infty)=0$. Find $T(i)$ and describe $T(R)$, where R is the real line.

I ended up getting $T(z)=\frac{1}{(-i-1)z+1}$, and I am pretty sure this is correct. I am lost on what $T(Z)$ does to the real line.

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    While you are at it, do $ T \left( \frac{1}{2} \right) $ and include that in the picture.2012-08-14

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As Will Jagy notes in the comments, $T$ takes a line to a line or a circle. Since $T$ takes $0,1,\infty$ to $1,i,0$, it takes the real line to a circle through $1$, $i$, and $0$. Think about that and you'll see we're talking about the circle that goes through the 4 corners of that little square, so it's centered at the center of that square, and its radius is half the diagonal of that square (in agreement with Tunococ's comment).