Representação gráfica de uma função em um referencial ortonormal bem escolhido

Question

Quero criar exatamente esta imagem em látex, mas não consigo encontrar o código para fazer isso.... Esta é minha tentativa:

\begin{tikzpicture}
\begin{axis}\[
    axis lines=middle,
    axis line style={-Latex, thick},
    xlabel={$x$},
    ylabel={$y$},
    xlabel style={at={(ticklabel* cs:1)},anchor=north west},
    ylabel style={at={(ticklabel* cs:1)},anchor=south west},
    xmin=-4, xmax=4,
    ymin=-4, ymax=4,
    xtick={-3,-2,...,3},
    ytick={-3,-2,...,3},
    grid=both,
    grid style={line width=.1pt, draw=gray!10},
    major grid style={line width=.2pt,draw=gray!50},
    minor tick num=1, % number of minor ticks between major ticks
    ticklabel style={font=\small,fill=white},
    legend style={draw=none, at={(0.5,-0.1)},anchor=north},
    legend cell align={left}
\]

\addplot \[
    domain=-3:3, 
    samples=100, 
    color=red,
    thick
\]
{1/x};
\addlegendentry{$y = \frac{1}{x}$}

% Add points
\node\[label={180:{(1,1)}},circle,fill,inner sep=2pt\] at (axis cs:1,1) {};
\node\[label={180:{(2,0.5)}},circle,fill,inner sep=2pt\] at (axis cs:2,0.5) {};
\node\[label={180:{(3,0.333)}},circle,fill,inner sep=2pt\] at (axis cs:3,0.333) {};

\end{axis}
\end{tikzpicture}]

Answer 1

Assim:

Código ( tikz):

\documentclass[a4paper]{article}
\usepackage{tikz}

\begin{document}
    
    \begin{center}
        \begin{tikzpicture}[font=\sffamily\small]
            %
            \draw[gray!20,step=0.2] (-4,-4) grid (4,4);
            \draw[gray!40] (-4,-4) grid (4,4);
            %
            \draw[->,thick] (-4,0) -- (4.5,0) node[anchor=west]{$x$};
            \draw[->,thick] (0,-4) -- (0,4.5) node[anchor=south]{$y$};
            %
            \foreach \x in {-4,...,-1,1,2,3,4} \draw [thick](\x cm,-2pt) -- (\x cm,2pt) node[below] {\x};
            \foreach \y in {-4,...,-1,1,2,3,4} \draw [thick](-2pt,\y) -- (2pt,\y) node[left]{\y};
            \node at (-.2,-.2) () {$O$};
            %
            \clip (-4,-4) rectangle (4,4);
            \draw[red,line width=2pt,step=.01] plot[domain=-4:-.1,smooth,samples=50] (\x,{1/\x});
            \draw[red,line width=2pt,step=.01] plot[domain=.1:4,smooth,samples=50] (\x,{1/\x});
            \fill (1,1) circle(2pt) node[above right] () {\bfseries $y=1$};
            \fill (2,.5) circle(2pt) node[above right] () {\bfseries $y=\frac12$};
            \fill (3,.33) circle(2pt) node[above right] () {\bfseries $y=\frac13$};
        \end{tikzpicture}
    \end{center}
\end{document}

Note: divido o domínio em duas partes para pular a singularidade em x=0.