Let (C=left{(x, y) in mathbb{R}^{2} mid x^{2}+y^{2} leq 1 ight}), and consider the following functions (g, h,

Let \(C=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\right\}\), and consider the following functions \(g, h, f: C \rightarrow C:\)

Let \(G=g(C)=\{g(x, y) \mid(x, y) \in C\}\). Describe the geometric shape of \(C\) and \(G\), and how the functions \(g\) and \(h\) act. Does each of the functions \(g, h, f\) have a fixed point in C? Justify why, or why not. Determine all fixed points, if any, of each of \(g, h\), and \(f\).

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g(x, y) = (x, V1-x), h(x, y) = (-y,x), f(x, y) = h(g(x, y)).