本题目来源于试卷: Exponent-Log Functions，类别为 IB数学
[问答题]
Consider $ f(x)=\frac{1}{2}-\ln \left(\sqrt{x^{2}-4}\right) $.
1. Find the largest possible domain D for f to be a function.
The function f is defined by $f(x)=\frac{1}{2}-\ln \left(\sqrt{x^{2}-4}\right)$, for $x \in D $.
2. Sketch the graph of y=f(x) , showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
3. Explain why f is an even function.
4. Explain why the inverse function $f^{-1}$ does not exist.
The function g is defined by $g(x)=\frac{1}{2}-\ln \left(\sqrt{x^{2}-4}\right)$ , for $x \in(2, \infty)$ .
5. Find the inverse function $g^{-1}$ and state its domain.
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