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> Let $a$ and $b$ be some complex numbers and let be $f: \mathbb{C} \to \mathbb{C}$ such that $f(z)=az + b\overline{z}$. Prove that $f$ is bijective if and only if $|a|-|b|\neq 0$.
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My idea was to find another function $g: \mathbb{C} \to \mathbb{C}$ with the property $g \circ f = f \circ g= \mathbb{1}\_{\mathbb{C}}$ and to use the condition $|a| \neq |b| \Leftrightarrow |a|^2 \neq |b|^2$ but I do not know how to continue. Could somebody help me?