How to make the unit length of $x$ axis and that of $y$ axis the same?

Question

When I use Tikz to draw the graphs of $y=x+\frac{1}{x}$ and $y=5(x-\frac{4}{3})$, but I found the unit length of $x$ axis and that of $y$ axis are different, if I want to let them be the same, what should I do?

Answer 1

You can enforce equal units by saying unit vector ratio=1 1 1, axis equal image or axis equal. The differences are subtle.

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usepackage{pdflscape}
\begin{document}
\begin{landscape}
\begin{tikzpicture}[
    declare function={
    f(\x) = \x+1/\x;
    g(\x) = 5*(\x-4/3);}] 
\begin{axis}[unit vector ratio=1 1 1,
    axis x line=middle, axis y line=middle,
    ymin=-5, ymax=5,ylabel=$y$, 
    xmin=0,xmax=5.5,xlabel=$x$,
    domain=0.1:5,samples=101,
    legend pos=south east]
\addplot [blue,thick] {f(x)};
\addlegendentry{$f(x)$}
\addplot [red,thick] {g(x)};
\addlegendentry{$g(x)$}
\end{axis}
\node[anchor=north] at (current axis.south) {\texttt{unit vector ratio=1 1 1}};
\end{tikzpicture} 
~
\begin{tikzpicture}[
    declare function={
    f(\x) = \x+1/\x;
    g(\x) = 5*(\x-4/3);}] 
\begin{axis}[axis equal image,
    axis x line=middle, axis y line=middle,
    ymin=-5, ymax=5,ylabel=$y$, 
    xmin=0,xmax=5.5,xlabel=$x$,
    domain=0.1:5,samples=101,
    legend pos=south east]
\addplot [blue,thick] {f(x)};
\addlegendentry{$f(x)$}
\addplot [red,thick] {g(x)};
\addlegendentry{$g(x)$}
\end{axis}
\node[anchor=north] at (current axis.south) {\texttt{axis equal image}};
\end{tikzpicture} 
~
\begin{tikzpicture}[
    declare function={
    f(\x) = \x+1/\x;
    g(\x) = 5*(\x-4/3);}] 
\begin{axis}[axis equal,
    axis x line=middle, axis y line=middle,
    ymin=-5, ymax=5,ylabel=$y$, 
    xmin=0,xmax=5.5,xlabel=$x$,
    domain=0.1:5,samples=101,
    legend pos=south east]
\addplot [blue,thick] {f(x)};
\addlegendentry{$f(x)$}
\addplot [red,thick] {g(x)};
\addlegendentry{$g(x)$}
\end{axis}
\node[anchor=north] at (current axis.south) {\texttt{axis equal}};
\end{tikzpicture} 
\end{landscape}
\end{document}