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Suppose that \(f\) has a fixed point at \(z_0\text{.}\) Show that the function \(g\) obtained by conjugating \(f\) with the function \(\varphi(z)=z+z_0\) has a fixed point at zero. In addition, show that the conjugation preserves the nature of the fixed point as attractive, super-attractive, repulsive, or neutral.