How to limit the range on a function in a TiKZ

Question

How do I limit the y values on the following tikz picture, so that the functions are contained in the box? I tried 'clipping' it into a rectangle, but this makes it hard to format it, and to write captions around the figure. Any tips?

\begin{figure}[h]
\centering
\begin{minipage}{0.48\textwidth}
\begin{tikzpicture}[scale=1.3]
\foreach \a/\Col in {0.25/green,3/red, 1/black}
{
\draw[\Col] plot[domain=0:4, range=0:4,variable=\x,samples=90] ({\x},{0.625*(\x^(\a+1))/(0.5*(\x^\a+1))});
}
\foreach \a/\Col\dashing in {3/cyan/dashed,0.25/magenta/dashed, 1/black}
{
\draw[\Col, \dashing] plot[domain=0:4, range=0:4, variable=\x,samples=90] ({\x},{\a*(\x^2)/(0.5*(1+\a*\x)});
}
\draw (0,0) rectangle (4,4);
\draw [dotted] (1,0) node[below]{$1$} -- (1,4);
\draw [dotted] (0,1.1) node[left]{$0.5$} -- (4,1.1);
\node at (3.6,-.2) {$v_i$};
\node at (-.2,3.85) {\small MRS};
\node [text=red, rotate=26] at (2.3,3.25) {\small $\alpha=4$};
\node [text=green, rotate=28] at (2.9,2.1) {\small $\alpha=1/4$};
\node [text=cyan, rotate=13] at (1.55,1.42) {\small $a_i=4$};
\node [text=magenta, rotate=10] at (0.85,0.2) {\small $a_i=4$};
\end{tikzpicture}
\end{minipage}
\end{figure}

Answer 1

@Zarko has already shown a very nice way to do this with pgfplots. Here is a slightly more manual way to do it with pgfplots which involves plotting each line separately. It's not too tedious in fairness, and allows you easy control of the different colours, line styles and, in particular, the positioning of these custom labels which you want to run parallel to the lines I presume?

\documentclass[margin=0.5cm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}    

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    xlabel={$v_i$},
    ylabel={MRS},
    every axis x label/.style={at={(current axis.right of origin)},below},
    every axis y label/.style={at={(current axis.north west)},left},
    xtick={1},
    ytick={0.5},
    xtick style={draw=none},
    ytick style={draw=none},
    no marks,
    xmin=0,xmax=4, 
    ymin=0,ymax=4,
    domain=0:4,
    samples=90
    ]

\draw [dotted] (axis cs:1,0) -- (axis cs:1,4);
\draw [dotted] (axis cs:0,0.5) -- (axis cs:4,0.5);

\newcommand{\aVal}{0.25}
\addplot [green] {0.625*(\x^(\aVal+1))/(0.5*(\x^\aVal+1))} node[green,pos=0.6,sloped,anchor=north] {$\alpha = 1/4$};
\renewcommand{\aVal}{3}
\addplot [red] {0.625*(\x^(\aVal+1))/(0.5*(\x^\aVal+1))} node[red,pos=0.6,sloped,anchor=south] {$\alpha = 4$};
\renewcommand{\aVal}{1}
\addplot [black] {0.625*(\x^(\aVal+1))/(0.5*(\x^\aVal+1))};

\newcommand{\bVal}{3}
\addplot [cyan,dashed] {\bVal*(x^2)/(0.5*(1+\bVal*x)} node[cyan,pos=0.4,sloped,anchor=south] {$a_i = 4$};
\renewcommand{\bVal}{0.25}
\addplot [magenta,dashed] {\bVal*(x^2)/(0.5*(1+\bVal*x)} node[magenta,pos=0.2,sloped,anchor=north] {$a_i = 4$};
\renewcommand{\bVal}{1}
\addplot [black] {\bVal*(x^2)/(0.5*(1+\bVal*x)};

\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

with pgfplots is simple ... with small redesign of your diagram you can obtain:

\documentclass[margin=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
grid,
xmax=4, ymax=4,
enlargelimits=false,
legend pos=north west,
legend cell align=left,
domain=0:4,
samples=90,
no marks
            ]
\foreach \a in {0.25,3, 1}
{
\addplot  {0.625*(x^(\a+1))/(0.5*(x^\a+1))};
}
\foreach \b in {3,0.25, 1}
{
\addplot +[dashed] {\b*(x^2)/(0.5*(1+\b*x)};
}
\legend{a=0.25, a=3, a=1,
        b=3, b=0.25, b=1}
\end{axis}
\end{tikzpicture}
\end{document}

addendum: to draw only dashed grid lines as shown on provided picture you can use extra x tick and extra y tick:

\documentclass[margin=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
grid,
xtick=\empty,ytick=\empty,
extra x ticks={1,4}, extra y ticks={1,4},
extra x tick labels={1,$v_i$},
extra y tick labels={0.5, RMS},
extra tick style={dashed},
xmax=4, ymax=4,
enlargelimits=false,
legend pos=north west,
legend cell align=left,
legend style={font=\footnotesize,draw=none},
domain=0:4,
samples=90,
no marks
            ]
\foreach \a in {0.25,3, 1}
{
\addplot  {0.625*(x^(\a+1))/(0.5*(x^\a+1))};
}
\foreach \b in {3,0.25, 1}
{
\addplot +[dashed] {\b*(x^2)/(0.5*(1+\b*x)};
}
\legend{$\alpha=0.25$, $\alpha=3$, $\alpha=1$,
        $a_i=3$, $a_1=0.25$, $a_1=1$}
\end{axis}
\end{tikzpicture}
\end{document}

in both cases i suggest to use legend. In my opinion, the battle clearly indicates the parameters of the curves.

Answer 3

Without pgfplots but \clip instead and a style for sloped labels, i.e. you don't have to put the rotation angles nor positions by hand.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,calc}
\begin{document}

\tikzset{annotate/.style args={#1,#2,#3}{postaction={decorate,decoration={markings,%
mark=at position #1-0.1 with {\coordinate (Xpfft1);},
mark=at position #1 with {\coordinate (Xpfft2);
\pgftransformreset
\path let \p1=($(Xpfft2)-(Xpfft1)$) in
\pgfextra{\pgfmathsetmacro{\myangle}{atan2(\y1,\x1)}\xdef\myangle{\myangle}};
\path (Xpfft1) -- (Xpfft2) node[rotate=\myangle,above,#2] {#3};
}}}}}
\begin{figure}[h]
\centering
\begin{minipage}{0.48\textwidth}
\begin{tikzpicture}[scale=1.3]
\draw (0,0) rectangle (4,4);
\clip (-1,-1) rectangle (4,4);
\foreach \a/\Col in {0.25/green,3/red, 1/black}
{
\draw[\Col,annotate={0.7,{text=\Col,font=\small},$\alpha=\a$}] 
plot[domain=0:4, range=0:4,variable=\x,samples=90] ({\x},{0.625*(\x^(\a+1))/(0.5*(\x^\a+1))});
}
\foreach \a/\Col\dashing in {3/cyan/dashed,0.25/magenta/dashed, 1/black}
{
\draw[\Col, \dashing,annotate={0.4,{text=\Col,font=\small},$\alpha=\a$}] plot[domain=0:4, range=0:4, variable=\x,samples=90] ({\x},{\a*(\x^2)/(0.5*(1+\a*\x)});
}
\draw [dotted] (1,0) node[below]{$1$} -- (1,4);
\draw [dotted] (0,1.1) node[left]{$0.5$} -- (4,1.1);
\node at (3.6,-.2) {$v_i$};
\node [font=\small]at (-.2,3.85) {MRS};
\end{tikzpicture}
\end{minipage}
\end{figure}
\end{document}