Considere as funções

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

­

\[{f + g}\]

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

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

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

\[{f \times g}\]

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

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

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

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

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

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

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

\[{h – g}\]

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

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

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

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

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

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

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