2.16:
a: ĐKXĐ: \(\left\{{}\begin{matrix}x>=0\\\sqrt{x}-1< >0\end{matrix}\right.\)
=>x>=0 và x<>1
b: \(f\left(4-2\sqrt{3}\right)=\dfrac{\sqrt{4-2\sqrt{3}}+1}{\sqrt{4-2\sqrt{3}}-1}\)
\(=\dfrac{\sqrt{\left(\sqrt{3}-1\right)^2}+1}{\sqrt{\left(\sqrt{3}-1\right)^2}-1}=\dfrac{\sqrt{3}-1+1}{\sqrt{3}-1-1}\)
\(=\dfrac{\sqrt{3}}{-2+\sqrt{3}}=-\dfrac{\sqrt{3}}{2-\sqrt{3}}=-\sqrt{3}\left(2+\sqrt{3}\right)=-2\sqrt{3}-3\)
\(f\left(a^2\right)=\dfrac{\sqrt{a^2}+1}{\sqrt{a^2}-1}=\dfrac{-a+1}{-a-1}=\dfrac{a-1}{a+1}\)
c: \(f\left(x\right)=\sqrt{3}\)
=>\(\sqrt{x}+1=\sqrt{3}\cdot\sqrt{x}-\sqrt{3}\)
=>\(\sqrt{x}\left(1-\sqrt{3}\right)=-\sqrt{3}-1\)
=>\(\sqrt{x}\left(\sqrt{3}-1\right)=\sqrt{3}+1\)
=>\(\sqrt{x}=\dfrac{\sqrt{3}+1}{\sqrt{3}-1}\)
=>\(x=\dfrac{4+2\sqrt{3}}{4-2\sqrt{3}}=\dfrac{2+\sqrt{3}}{2-\sqrt{3}}=\left(2+\sqrt{3}\right)^2=7+4\sqrt{3}\)
d: f(x)=f(x^2)
=>\(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}=\dfrac{\left|x\right|+1}{\left|x\right|-1}\)
=>\(\left(\sqrt{x}+1\right)\left(\left|x\right|-1\right)=\left(\sqrt{x}-1\right)\left(\left|x\right|+1\right)\)
=>\(\sqrt{x}\cdot\left|x\right|-\sqrt{x}+\left|x\right|-1=\sqrt{x}\cdot\left|x\right|+\sqrt{x}-\left|x\right|-1\)
=>\(-2\sqrt{x}+2\left|x\right|=0\)
=>\(\left[{}\begin{matrix}x=0\left(nhận\right)\\x=1\left(loại\right)\end{matrix}\right.\)
