本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
A function f(x) is ;b.g /qdcpca j(;g;ul defigz2.1f+cu/ zmh(/ lqfi ky(uened by $ f(x)=\arccos \left(\frac{x^{2}-1}{x^{2}+1}\right), x \in \mathbb{R}$ .
1. Show that f is an even function.
2. Find the equation of the horizontal asymptote to the graph of y=f(x) .
3. 1. Show that $f^{\prime}(x)=-\frac{2 x}{\sqrt{x^{2}}\left(x^{2}+1\right)} $ for $x \in \mathbb{R},$ $x \neq 0 $.
2. Using the expression for $f^{\prime}(x)$ and the result $\sqrt{x^{2}}=|x|$ , show that f is increasing for $x\lt 0$ .
A function g is defined by $ g(x)=\arccos \left(\frac{x^{2}-1}{x^{2}+1}\right), x \in \mathbb{R}$, $x \geq 0 $.
4. Find the range of g .
5. Find an expression for $g^{-1}(x)$ .
6. State the domain of $g^{-1}(x)$ .
7. Sketch the graph of $y=g^{-1}(x)$ . Clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.
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