Duas funções reais de variável real

Considere as funções reais de variável real $f$ e $g$ definidas por $$\begin{matrix}
f(x)={{e}^{2x+1}} & {} & {} & g(x)=\ln \left( 3-3x \right)  \\
\end{matrix}$$

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1.  Os domínios das funções $f$ e $g$ são:
   $$\begin{array}{*{35}{l}}
   {{D}_{f}} & = & \left\{ x\in \mathbb{R}:\left( 2x+1 \right)\in \mathbb{R} \right\}  \\
   {} & = & \mathbb{R}  \\
   \end{array}$$
   \[\begin{array}{*{35}{l}}
   {{D}_{g}} & = & \left\{ x\in \mathbb{R}:3-3x>0 \right\}  \\
   {} & = & \left] -\infty ,1 \right[  \\
   \end{array}\]
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2. Comecemos por determinar o domínio de $f\circ g$:\[\begin{array}{*{35}{l}}
   {{D}_{f\circ g}} & = & \left\{ x\in \mathbb{R}:x\in {{D}_{g}}\wedge g(x)\in {{D}_{f}} \right\}  \\
   {} & = & \left\{ x\in \mathbb{R}:x<1\wedge (\ln \left( 3-3x \right))\in \mathbb{R} \right\}  \\
   {} & = & \left\{ x\in \mathbb{R}:x<1\wedge x<1 \right\}  \\
   {} & = & \left] -\infty ,1 \right[  \\
   \end{array}\]
   Ora, $$\begin{array}{*{35}{l}}
   (f\circ g)(x) & = & f(g(x))  \\
   {} & = & f(\ln \left( 3-3x \right))  \\
   {} & = & {{e}^{2\times (\ln \left( 3-3x \right))+1}}  \\
   {} & = & {{e}^{2\times \ln \left( 3-3x \right)}}\times {{e}^{1}}  \\
   {} & = & {{e}^{1}}\times {{e}^{\ln {{\left( 3-3x \right)}^{2}}}}  \\
   {} & = & e\times {{(3-3x)}^{2}}  \\
   \end{array}$$
   Logo: \[\begin{array}{*{35}{l}}
   f\circ g: & \left] -\infty ,1 \right[\to \mathbb{R}  \\
   {} & x\to e\times {{(3-3x)}^{2}}  \\
   \end{array}\]