Question

The rational function \(c(x) := \frac{x+1}{x-1}\) is called Cayley transformation
b) Show that \(c:U\rightarrow V\) is bijective

Answer 1

Are \(U, V \subseteq \mathbb{R}\)?

Answer 2

hmm thats not given but above its written there rational function c(x):...
i think its U, V \subset \mathbb{R}

Answer 3

also i mean we suppose yes..

Answer 4

understand?

Answer 5

First show that \(c(x)\) is one-to-one.
\(c(x)\) is one-to-one if for all \(u_1, u_2\in U, c(u_1)=c(u_2) \Rightarrow u_1=u_2\).
Then show that \(c(x)\) is onto.
\(c(x)\) is onto if for all \(v\in V, \exists u\in U(c(u)=v)\).

Answer 6

ok thank you blockfolder i have 2 more smilar question, i will post it very soon. i would be happy if you take look at it too