#title: Domínio de função

#author: Smirnov




#let:

n a = [1,2,3,4];
n b = [-1,1,-2,2,-3,3,-4,4];
n c = [-1,1,-2,2,-3,3,-4,4];


f F1 = [sqrt(z), sqrt(-z), log(z), log(-z), 1/z, 1/sqrt(z), 1/sqrt(-z)];

s S = [\geq, \leq, >, <, \neq, >, <];



f D0  = [ #a^2-(x+#b)^2-(y+#c)^2];
f D1 = ratsimp(#D0);


f d = ratsimp(-#b);

f e = ratsimp(-#c);


f aux1 = [F(z) := #F1];

f G  = ratsimp(F(#D1));


fig1 = {{ \begin{tikzpicture}[scale=0.5]
\fill[blue!20] (#d,#e) circle (#a);
\draw[->](-8,0)--(8,0) node[right]{$x$};
\draw[->](0,-8)--(0,8) node[above]{$y$};
\node at (#d,#e) [below] {$(#d,#e)$};
\filldraw[black] (#d,#e) circle(1.5pt);
\foreach \x/\xtext in {-6/6, -4/4, -2/2, 2/2,  4/4,  6/6, }
\draw[shift={(\x,0)}] (0pt,0.3pt) -- (0pt,-0.3pt) node[below] {$\xtext$};
\foreach \y/\ytext in {-6/6, -4/4, -2/2, 2/2, 4/4, 6/6}
\draw[shift={(0,\y)}] (0.3pt,0pt) -- (-0.3pt,0pt) node[left] {$\ytext$};
\foreach \x in {-7,...,7}
\draw (\x cm,-1pt) -- (\x cm,1pt);
\foreach \y in {-7,...,7}
\draw (-1pt,\y cm) -- (1pt,\y cm);
\end{tikzpicture} }};




fig2 = {{ \begin{tikzpicture}[scale=0.5]
\filldraw[blue!20] (-7,-7) rectangle (7,7);
\filldraw[white] (#d,#e) circle (#a);
\draw[->](-8,0)--(8,0) node[right]{$x$};
\draw[->](0,-8)--(0,8) node[above]{$y$};
\node at (#d,#e) [below] {$(#d,#e)$};
\filldraw[black] (#d,#e) circle(1.5pt);
\foreach \x/\xtext in {-6/6, -4/4, -2/2, 2/2,  4/4,  6/6, }
\draw[shift={(\x,0)}] (0pt,0.3pt) -- (0pt,-0.3pt) node[below] {$\xtext$};
\foreach \y/\ytext in {-6/6, -4/4, -2/2, 2/2, 4/4, 6/6}
\draw[shift={(0,\y)}] (0.3pt,0pt) -- (-0.3pt,0pt) node[left] {$\ytext$};
\foreach \x in {-7,...,7}
\draw (\x cm,-1pt) -- (\x cm,1pt);
\foreach \y in {-7,...,7}
\draw (-1pt,\y cm) -- (1pt,\y cm);
\end{tikzpicture} }};



fig3 = {{ \begin{tikzpicture}[scale=0.5]
\draw[style=dashed,color=black] (#d,#e) circle (#a);
\fill[blue!20] (#d,#e) circle (#a);
\draw[->](-8,0)--(8,0) node[right]{$x$};
\draw[->](0,-8)--(0,8) node[above]{$y$};
\node at (#d,#e) [below] {$(#d,#e)$};
\filldraw[black] (#d,#e) circle(1.5pt);
\foreach \x/\xtext in {-6/6, -4/4, -2/2, 2/2,  4/4,  6/6, }
\draw[shift={(\x,0)}] (0pt,0.3pt) -- (0pt,-0.3pt) node[below] {$\xtext$};
\foreach \y/\ytext in {-6/6, -4/4, -2/2, 2/2, 4/4, 6/6}
\draw[shift={(0,\y)}] (0.3pt,0pt) -- (-0.3pt,0pt) node[left] {$\ytext$};
\foreach \x in {-7,...,7}
\draw (\x cm,-1pt) -- (\x cm,1pt);
\foreach \y in {-7,...,7}
\draw (-1pt,\y cm) -- (1pt,\y cm);
\end{tikzpicture}  }};



fig4 = {{ \begin{tikzpicture}[scale=0.5]
\filldraw[blue!20] (-7,-7) rectangle (7,7);
\filldraw[white] (#d,#e) circle (#a);
\draw[style=dashed,color=black] (#d,#e) circle (#a);
\draw[->](-8,0)--(8,0) node[right]{$x$};
\draw[->](0,-8)--(0,8) node[above]{$y$};
\node at (#d,#e) [below] {$(#d,#e)$};
\filldraw[black] (#d,#e) circle(1.5pt);
\foreach \x/\xtext in {-6/6, -4/4, -2/2, 2/2,  4/4,  6/6, }
\draw[shift={(\x,0)}] (0pt,0.3pt) -- (0pt,-0.3pt) node[below] {$\xtext$};
\foreach \y/\ytext in {-6/6, -4/4, -2/2, 2/2, 4/4, 6/6}
\draw[shift={(0,\y)}] (0.3pt,0pt) -- (-0.3pt,0pt) node[left] {$\ytext$};
\foreach \x in {-7,...,7}
\draw (\x cm,-1pt) -- (\x cm,1pt);
\foreach \y in {-7,...,7}
\draw (-1pt,\y cm) -- (1pt,\y cm);
\end{tikzpicture} }};


fig5 = {{ \begin{tikzpicture}[scale=0.5]
\filldraw[blue!20] (-7,-7) rectangle (7,7);
\draw[style=dashed,color=white] (#d,#e) circle (#a);
\draw[->](-8,0)--(8,0) node[right]{$x$};
\draw[->](0,-8)--(0,8) node[above]{$y$};
\node at (#d,#e) [below] {$(#d,#e)$};
\filldraw[black] (#d,#e) circle(1.5pt);
\foreach \x/\xtext in {-6/6, -4/4, -2/2, 2/2,  4/4,  6/6, }
\draw[shift={(\x,0)}] (0pt,0.3pt) -- (0pt,-0.3pt) node[below] {$\xtext$};
\foreach \y/\ytext in {-6/6, -4/4, -2/2, 2/2, 4/4, 6/6}
\draw[shift={(0,\y)}] (0.3pt,0pt) -- (-0.3pt,0pt) node[left] {$\ytext$};
\foreach \x in {-7,...,7}
\draw (\x cm,-1pt) -- (\x cm,1pt);
\foreach \y in {-7,...,7}
\draw (-1pt,\y cm) -- (1pt,\y cm);
\end{tikzpicture} }};


fig6 = {{ \begin{tikzpicture}[scale=0.5]
\draw[style=dashed,color=black] (#d,#e) circle (#a);
\fill[blue!20] (#d,#e) circle (#a);
\draw[->](-8,0)--(8,0) node[right]{$x$};
\draw[->](0,-8)--(0,8) node[above]{$y$};
\node at (#d,#e) [below] {$(#d,#e)$};
\filldraw[black] (#d,#e) circle(1.5pt);
\foreach \x/\xtext in {-6/6, -4/4, -2/2, 2/2,  4/4,  6/6, }
\draw[shift={(\x,0)}] (0pt,0.3pt) -- (0pt,-0.3pt) node[below] {$\xtext$};
\foreach \y/\ytext in {-6/6, -4/4, -2/2, 2/2, 4/4, 6/6}
\draw[shift={(0,\y)}] (0.3pt,0pt) -- (-0.3pt,0pt) node[left] {$\ytext$};
\foreach \x in {-7,...,7}
\draw (\x cm,-1pt) -- (\x cm,1pt);
\foreach \y in {-7,...,7}
\draw (-1pt,\y cm) -- (1pt,\y cm);
\end{tikzpicture} }};


fig7 = {{ \begin{tikzpicture}[scale=0.5]
\filldraw[blue!20] (-7,-7) rectangle (7,7);
\filldraw[white] (#d,#e) circle (#a);
\draw[style=dashed,color=black] (#d,#e) circle (#a);
\draw[->](-8,0)--(8,0) node[right]{$x$};
\draw[->](0,-8)--(0,8) node[above]{$y$};
\node at (#d,#e) [below] {$(#d,#e)$};
\filldraw[black] (#d,#e) circle(1.5pt);
\foreach \x/\xtext in {-6/6, -4/4, -2/2, 2/2,  4/4,  6/6, }
\draw[shift={(\x,0)}] (0pt,0.3pt) -- (0pt,-0.3pt) node[below] {$\xtext$};
\foreach \y/\ytext in {-6/6, -4/4, -2/2, 2/2, 4/4, 6/6}
\draw[shift={(0,\y)}] (0.3pt,0pt) -- (-0.3pt,0pt) node[left] {$\ytext$};
\foreach \x in {-7,...,7}
\draw (\x cm,-1pt) -- (\x cm,1pt);
\foreach \y in {-7,...,7}
\draw (-1pt,\y cm) -- (1pt,\y cm);
\end{tikzpicture} }};



Gr = [#fig1, #fig2, #fig3, #fig4, #fig5, #fig6, #fig7];

F1~S~Gr;


#question:

\noindent Determine  e represente geometricamente o domínio da função $f(x,y)=#G$.


#sugestion:


#resolution:






#result:
$$
 {\rm dom} f = \{ (x,y) \mid  #D0 #S 0\}.
$$


$$
 #Gr
$$


A área azul representa o domínio da função.


#Verification:


#usepackage:
 
\usepackage{tikz}