Weird looking square root on pgfplots

Question

I'm trying to plot a simple sqrt(x) function and I get a weird looking one.

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}
    \begin{tikzpicture}
        \begin{axis} [
        smooth, no markers, grid,
        domain=0:2,
        xmax=2, ymax=2,
        xmin=0, ymin=0]

        \addplot +[red] {x^2};
        \addplot +[blue]{sqrt(x)};
        \addplot {x};
        \end{axis}
    \end{tikzpicture}
\end{document}

I'm plotting it together with x^2 and what I get is clearly not the inverse function:

Answer 1

You can see why this happens if you remove no markers from the axis options. A quick fix is to have denser sampling near zero, with help from samples at. Personally I'd also not use smooth, the results aren't always that good.

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
    \begin{axis} [
    no markers, grid,
    domain=0:2,
    xmax=2, ymax=2,
    xmin=0, ymin=0]

    \addplot +[red,samples=50] {x^2};
    \addplot +[blue,samples at={0,0.001,0.005,0.01,...,0.2,0.3,0.4,...,2}]{sqrt(x)};
    \addplot +[samples=2] {x};
    \end{axis}
\end{tikzpicture}
\end{document}

Answer 2

As Torbjørn states in his answer the key is to increase the sampling density around x=0. But to my personal view stating the x samples by hand (using samples at) is the most inelegant way.

The simplest (good) solution is given by marsupilam in the comment below Torbjørn's answer which he hopefully will transfer into a real answer.

Here I present a general way to get unequal spacing and the "inequality" can be changed by simple changing the constant a. Applying the same method on a much more complex equation can be found at https://tex.stackexchange.com/a/373820/95441.

% used PGFPlots v1.15
\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
    \pgfplotsset{
        % use this `compat' level or higher to use Lua for calculations
        % (this is not required though)
        compat=1.12,
        /pgf/declare function={
            % declare the main function(s)
            f(\x) = sqrt(\x);
            % state (or calculate) the lower and upper boundaries (the domain values)
            lb = 0;
            ub = 2;
            %
            % -----------------------------------------------------------------
            %%% nonlinear spacing: <https://stackoverflow.com/a/39140096/5776000>
            % "non-linearity factor"
            a = 2;
            % function to use for the nonlinear spacing
            Y(\x) = exp(a*\x);
            % rescale to former limits
            X(\x) = (Y(\x) - Y(lb))/(Y(ub) - Y(lb)) * (ub - lb) + lb;
        },
    }
\begin{document}
\begin{tikzpicture}
    \begin{axis}[
        % just to show that also here the above constants/functions can be used
        xmin=lb,    xmax=ub,
        ymin=f(lb), ymax=ub,
        grid,
        % also use the constants for the domain, so there is only one place
        % where you need to change the values
        domain=lb:ub,
        smooth,
        no markers,     % <-- comment me to show where the x points are
    ]
        \addplot+ [mark=o]              {x^2};
        \addplot+ [samples=2]           {x};
        \addplot+ [mark=triangle,thick] {f(x)};
        \addplot+ [mark=square,green]   ({X(x)},{f(X(x))});
    \end{axis}
\end{tikzpicture}
\end{document}

Answer 3

As requested by Stefan Pinnow, I expand my comment to an answer.

We just plot the original function x|->x^2 twice :

• one normally,
• and once after applying the x <-> y symetry of the plane. This is the graph of the reciprocal

The output is the same as other answers above.

Here, I changed the original function to showcase that it still works for non-bijective functions...

The output

The code

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
  [
    declare function= 
    {
      myFunction(\x) = .5*\x + sin(90 * \x);    % for demo
      %myFunction(\x) = \x^2;                   % your function
    },
  ]
  \begin{axis} 
    [
      unit vector ratio = 1 1,
      no markers,
      grid,
      domain=0:2,
      xmax=2,
      ymax=2,
      xmin=0,
      ymin=0,
      samples=50,
    ]

    \addplot +[red] {myFunction(x)}; % shortcut for ({x},{myFunction(x)});
    \addplot +[blue] ({myFunction(x)},{x});
    \addplot +[samples=2] {x};
  \end{axis}
\end{tikzpicture}
\end{document}