Caracterize as funções

\[\begin{array}{*{20}{l}}
{\begin{array}{*{20}{l}}
{f:}&{\mathbb{R}\backslash \left\{ 0 \right\} \to \mathbb{R}} \\
{}&{x \to \frac{1}{{{x^2}}}}
\end{array}}&{}&{}&{\begin{array}{*{20}{l}}
{g:}&{\mathbb{R}\backslash \left\{ { – 1} \right\} \to \mathbb{R}} \\
{}&{x \to x + 1}
\end{array}}&{}&{}&{\begin{array}{*{20}{l}}
{h:}&{\mathbb{R}\backslash \left\{ {0,1} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{1}{{{x^2} – x}}}
\end{array}}
\end{array}\]

\[{f + g}\]

\[{D_{f + g}} = {D_f} \cap {D_g} = \mathbb{R}\backslash \left\{ 0 \right\} \cap \mathbb{R}\backslash \left\{ { – 1} \right\} = \mathbb{R}\backslash \left\{ { – 1,0} \right\}\]

\[\left( {f + g} \right)\left( x \right) = f\left( x \right) + g\left( x \right) = \frac{1}{{{x^2}}} + x + 1 = \frac{{{x^3} + {x^2} + 1}}{{{x^2}}},\forall x \in {D_{f + g}}\]

\[\begin{array}{*{20}{l}}
{f + g:}&{\mathbb{R}\backslash \left\{ { – 1,0} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{{x^3} + {x^2} + 1}}{{{x^2}}}}
\end{array}\]

\[{f \times g}\]

\[{D_{f \times g}} = {D_f} \cap {D_g} = \mathbb{R}\backslash \left\{ 0 \right\} \cap \mathbb{R}\backslash \left\{ { – 1} \right\} = \mathbb{R}\backslash \left\{ { – 1,0} \right\}\]

\[\left( {f \times g} \right)\left( x \right) = f\left( x \right) \times g\left( x \right) = \frac{1}{{{x^2}}} \times \left( {x + 1} \right) = \frac{{x + 1}}{{{x^2}}},\forall x \in {D_{f \times g}}\]

\[\begin{array}{*{20}{l}}
{f \times g:}&{\mathbb{R}\backslash \left\{ { – 1,0} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{x + 1}}{{{x^2}}}}
\end{array}\]

\[{\frac{f}{g}}\]

\[{D_{\frac{f}{g}}} = {D_f} \cap {D_g} \cap \left\{ {x \in \mathbb{R}:g\left( x \right) \ne 0} \right\} = \mathbb{R}\backslash \left\{ 0 \right\} \cap \mathbb{R}\backslash \left\{ { – 1} \right\} \cap \mathbb{R}\backslash \left\{ { – 1} \right\} = \mathbb{R}\backslash \left\{ { – 1,0} \right\}\]

\[\left( {\frac{f}{g}} \right)\left( x \right) = \frac{{f\left( x \right)}}{{g\left( x \right)}} = \frac{{\frac{1}{{{x^2}}}}}{{x + 1}} = \frac{1}{{{x^3} + {x^2}}},\forall x \in {D_{\frac{f}{g}}}\]

\[\begin{array}{*{20}{l}}
{\frac{f}{g}:}&{\mathbb{R}\backslash \left\{ { – 1,0} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{1}{{{x^3} + {x^2}}}}
\end{array}\]

\[{h – g}\]

\[{D_{h – g}} = {D_h} \cap {D_g} = \mathbb{R}\backslash \left\{ {0,1} \right\} \cap \mathbb{R}\backslash \left\{ { – 1} \right\} = \mathbb{R}\backslash \left\{ { – 1,0,1} \right\}\]

\[\left( {h – g} \right)\left( x \right) = h\left( x \right) – g\left( x \right) = \frac{1}{{{x^2} – x}} – \left( {x + 1} \right) = \frac{{ – {x^3} – {x^2} + {x^2} + x + 1}}{{{x^2} – x}} = \frac{{ – {x^3} + x + 1}}{{{x^2} – x}},\forall x \in {D_{\frac{f}{g}}}\]

\[\begin{array}{*{20}{l}}
{h – g:}&{\mathbb{R}\backslash \left\{ { – 1,0,1} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{ – {x^3} + x + 1}}{{{x^2} – x}}}
\end{array}\]

\[\frac{f}{h}\]

\[{D_{\frac{f}{h}}} = {D_f} \cap {D_h} \cap \left\{ {x \in \mathbb{R}:h\left( x \right) \ne 0} \right\} = \mathbb{R}\backslash \left\{ 0 \right\} \cap \mathbb{R}\backslash \left\{ {0,1} \right\} \cap \mathbb{R}\backslash \left\{ {0,1} \right\} = \mathbb{R}\backslash \left\{ {0,1} \right\}\]

\[\left( {\frac{f}{h}} \right)\left( x \right) = \frac{{f\left( x \right)}}{{h\left( x \right)}} = \frac{{\frac{1}{{{x^2}}}}}{{\frac{1}{{{x^2} – x}}}} = \frac{{{x^2} – x}}{{{x^2}}} = \frac{{x\left( {x – 1} \right)}}{{{x^2}}} = \frac{{x – 1}}{x},\forall x \in {D_{\frac{f}{g}}}\]

\[\begin{array}{*{20}{l}}
{\frac{f}{h}:}&{\mathbb{R}\backslash \left\{ {0,1} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{x – 1}}{x}}
\end{array}\]

\[f \circ g\]

\[{D_{f \circ g}} = \left\{ {x \in \mathbb{R}:x \in {D_g} \wedge g\left( x \right) \in {D_f}} \right\} = \left\{ {x \in \mathbb{R}:x \in \mathbb{R}\backslash \left\{ { – 1} \right\} \wedge \left( {x + 1} \right) \in \mathbb{R}\backslash \left\{ 0 \right\}} \right\} = \left\{ {x \in \mathbb{R}:x \in \mathbb{R}\backslash \left\{ { – 1} \right\} \wedge x \in \mathbb{R}\backslash \left\{ { – 1} \right\}} \right\} = \mathbb{R}\backslash \left\{ { – 1} \right\}\]

\[\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right) = f\left( {x + 1} \right) = \frac{1}{{{{\left( {x + 1} \right)}^2}}},\forall x \in {D_{f \circ g}}\]

\[\begin{array}{*{20}{l}}
{f \circ g:}&{\mathbb{R}\backslash \left\{ { – 1} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{1}{{{{\left( {x + 1} \right)}^2}}}}
\end{array}\]

\[g \circ f\]

\[{D_{g \circ f}} = \left\{ {x \in \mathbb{R}:x \in {D_f} \wedge f\left( x \right) \in {D_g}} \right\} = \left\{ {x \in \mathbb{R}:x \in \mathbb{R}\backslash \left\{ 0 \right\} \wedge \left( {\frac{1}{{{x^2}}}} \right) \in \mathbb{R}\backslash \left\{ { – 1} \right\}} \right\} = \left\{ {x \in \mathbb{R}:x \in \mathbb{R}\backslash \left\{ 0 \right\} \wedge x \in \mathbb{R}\backslash \left\{ 0 \right\}} \right\} = \mathbb{R}\backslash \left\{ 0 \right\}\]

\[\left( {g \circ f} \right)\left( x \right) = g\left( {f\left( x \right)} \right) = g\left( {\frac{1}{{{x^2}}}} \right) = \frac{1}{{{x^2}}} + 1 = \frac{{{x^2} + 1}}{{{x^2}}},\forall x \in {D_{f \circ g}}\]

\[\begin{array}{*{20}{l}}
{g \circ f:}&{\mathbb{R}\backslash \left\{ 0 \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{{x^2} + 1}}{{{x^2}}}}
\end{array}\]