a) Mamy

$h\left(x\right)=f\left(g\left(x\right)\right)={\left(2x-1\right)}^2-1$  

$D_h=\mathbb{R}$  

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b) Mamy

$h\left(x\right)=f\left(g\left(x\right)\right)=\frac{1}{x^2+4x+4}=\frac{1}{{\left(x+2\right)}^2}$  

$D_h=\mathbb{R}\backslash \left\{-2\right\}$  

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c) Mamy

$h\left(x\right)=f\left(g\left(x\right)\right)=\sqrt{x^2+\left|x\right|}-x^2-\left|x\right|$  

$D_h=\mathbb{R}$  

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d) Mamy

$h\left(x\right)=f\left(g\left(x\right)\right)=\frac{x-1}{x+1}+\frac{x+1}{x-1}$  

$D_h=\mathbb{R}\backslash \left\{-1,1\right\}$