Fancy curves in Tikz

Using pgfplots:

\documentclass{article}

\usepackage[svgnames]{xcolor}
\usepackage{pgfplots}

\begin{document}

\def\b{1}

\begin{tikzpicture}
  \begin{axis}[%
    domain=0:4,
    samples=200,
    xmin=0,
    xmax=4,
    extra x ticks={1},
    extra x tick labels={},
    extra tick style={grid=major},
    xlabel={$b_1/b_2$},
    ymin=0,
    ymax=1,
    ytick={0,0.5,1},
    extra y ticks={0.5},
    extra y tick labels={},
    extra tick style={grid=major},
    ylabel={$x_1(b_1/b_2)$},
    ]
    \foreach \a/\clr in {0.25/DarkOliveGreen,
                          0.5/NavyBlue,
                            1/FireBrick,
                            2/DarkGoldenrod,
                            4/MediumSeaGreen}{%
      \expandafter\addplot\expandafter[\clr,
        line width=1pt,
        mark=none,
      ] {\a*((x/\b)^\a)/(\a*((x/\b)^\a) + \a*\b^\a)};
    }
  \end{axis}
\end{tikzpicture}

\end{document}

---

Ooh, too late :P

You want TikZ, right? (I mean no pgfplots.) So here we go.

\documentclass[tikz,border=5pt]{standalone}
\begin{document}
\begin{tikzpicture}[scale=2]
\foreach \a/\Col in {0.25/blue,0.5/red,1/orange,2/purple,4/black}
{
\draw[\Col] plot[domain=0:4,variable=\x,samples=90] ({\x},{4*(\a*\x^\a)/(\a + \a*\x^\a)});
}
\draw (0,0) rectangle (4,4);
\draw [dotted] (1,0) node[below]{$1$} -- (1,4);
\draw [dotted] (0,2) node[left](p5){$0.5$} -- (4,2);
\node [left of=p5,rotate=90]{$x_1(b_1/b_2)$};
\end{tikzpicture}
\end{document}

Just for fun: pgfplots. Phelype was faster with that.

\documentclass[tikz,border=2mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}
\begin{document}
\tikzset{declare function={f(\x,\y)=(\y*\x^\y)/(\y + \y*\x^\y);}}
\begin{tikzpicture}

\begin{axis}[domain=0:4,ytick={0,0.5,1},xtick={0,1,2,3,4},
xmin=0,ymin=0,xmax=4,ymax=1,mark=none,samples=100] 
\foreach \a in {0.25,0.5,1,2,4}
{ 
  \addplot[very thick] {f(x,\a)};
}  
\draw[dotted] (1,0) -- (1,1);
\draw[dotted] (0,0.5) -- (4,0.5);
\end{axis}  
\end{tikzpicture}

\end{document}