\(A=2\left|x\right|-4x+1\) \(\forall x\ge0\) A=\(2x-4x+1=1-2x\)
\(\forall x< 0\) A=\(-2x-4x+1=1-6x\)
B=\(\sqrt{x-1-2\sqrt{x-1}+1}=\sqrt{\left(x-1-1\right)^2}=\sqrt{\left(x-2\right)^2}=\left|x-2\right|\)
\(\forall x\ge2\) B = x-2 \(\forall x< 2\) B = 2-x
C=\(\sqrt{x-3+2.3\sqrt{x-3}+9}=\sqrt{\left(x-3+3\right)^2}=\left|x\right|\)
\(\forall x\ge0\) C=x \(\forall x< 0\) C=-x
\(a.A=\sqrt{4x^2}-4x+1=|2x|-4x+1\)
\(b.B=\sqrt{x-2\sqrt{x-1}}=\sqrt{x-1-2\sqrt{x-1}+1}=\sqrt{\left(\sqrt{x-1}-1\right)^2}=|\sqrt{x-1}+1|=\sqrt{x-1}+1\)
\(c.C=\sqrt{x+6+6\sqrt{x-3}}=\sqrt{x-3+6\sqrt{x-3}+9}=\sqrt{\left(\sqrt{x-3}+3\right)^2}=|\sqrt{x-3}+3|=\sqrt{x-3}+3\left(x\ge3\right)\)
\(d.D=\sqrt{x+2}+\dfrac{1}{\sqrt{x^2+2x+1}}=\sqrt{x+2}+\dfrac{1}{\sqrt{\left(x+1\right)^2}}=\sqrt{x+2}+\dfrac{1}{|x+1|}=\sqrt{x+2}+\dfrac{1}{x+1}\left(x\ge-2\right)\)
