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\documentclass[varwidth=false, border=2pt]{standalone}

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\usepackage{pgfplots}

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\usepackage{tikz}

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\usepackage{xcolor}

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\begin{document}

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\begin{tikzpicture}

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    \definecolor{color1}{HTML}{332288}

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    \definecolor{color2}{HTML}{FDB863}

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    \definecolor{color3}{HTML}{B2ABD2}

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    \definecolor{color4}{HTML}{5E3C99}

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    \begin{axis}[

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        legend pos=north west,

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        legend cell align={left},

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        axis x line=middle,

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        axis y line=middle,

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        x tick label style={/pgf/number format/fixed,

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                            /pgf/number format/fixed zerofill,

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                            /pgf/number format/precision=1},

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        y tick label style={/pgf/number format/fixed,

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                            /pgf/number format/fixed zerofill,

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                            /pgf/number format/precision=1},

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        grid = major,

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        width=16cm,

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        height=8cm,

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        grid style={dashed, gray!30},

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        xmin=-2,     % start the diagram at this x-coordinate

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        xmax= 2,    % end   the diagram at this x-coordinate

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        ymin=-1,     % start the diagram at this y-coordinate

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        ymax= 2,   % end   the diagram at this y-coordinate

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        %axis background/.style={fill=white},

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        xlabel=x,

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        ylabel=y,

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        tick align=outside,

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        enlargelimits=false]

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      % plot the stirling-formulae

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      \addplot[domain=-2:2, color1, ultra thick,samples=500] {1/(1+exp(-x))};

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      \addplot[domain=-2:2, color2, ultra thick,samples=500] {tanh(x)};

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      \addplot[domain=-2:2, color4, ultra thick,samples=500] {max(0, x)};

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      \addplot[domain=-2:2, color4, ultra thick,samples=500, dashed] {ln(exp(x) + 1)};

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      \addplot[domain=-2:2, color3, ultra thick,samples=500, dotted] {max(x, exp(x) - 1)};

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      \addlegendentry{$\varphi_1(x)=\frac{1}{1+e^{-x}}$}

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      \addlegendentry{$\varphi_2(x)=\tanh(x)$}

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      \addlegendentry{$\varphi_3(x)=\max(0, x)$}

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      \addlegendentry{$\varphi_4(x)=\log(e^x + 1)$}

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      \addlegendentry{$\varphi_5(x)=\max(x, e^x - 1)$}

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    \end{axis}

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\end{tikzpicture}

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\end{document}

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