Using Tikz to Sketch an irregular graph

I am trying to use tikz to sketch these graphs but it looks irregular and don't know how to go about it. I can do almost everything with the exception of the irregular graph in red which I would need help with. Any help would be very much appreciated.Thank you. \documentclass{article}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{decorations.shapes}
\usetikzlibrary{chains,arrows,positioning,decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{calc,fit}

\begin{document}

\begin{tikzpicture}
\begin{axis}[width=15cm,compat=1.5,height=9.5cm,
axis y line=center,
axis x line=middle,
xmin=-5,xmax=5,
ymin=-5,ymax=5,
xlabel=$x$,ylabel=$y$,
anchor=center
]
{(0,0) (1,-1) (4,-2)}; % node[right] {$y=\sqrt{x}$};

thick,mark=none,samples=200,unbounded coords=jump] {sqrt(x)};
\end{axis}
\end{tikzpicture}

\end{document}
• Please, show to us, what you do so far. Your code will be easy to complete with missing curve. Jun 26 '16 at 23:31
• I reformat your MWE and correct spelling error in it. However in it is missing, as can I see, package pgfplots (I didn't add it, since your MWE is in contradiction with your question). Now it is not very clear, what is your problem. Your function? If for it you don't know math function, you can determine it with coordinates and draw smooth curve through points. is this your problem? Jun 26 '16 at 23:55
• Thanks but I only wanted how I could plot the red graph in the picture using one of my old code for \sqrt{x} graph. Please help. Jun 27 '16 at 0:14
• See, if my answer fit to your wishes. Let me note: i don't understand what you mean with irregular graph, so I use two ordinary function and draw their graphs in selected domain. Jun 27 '16 at 2:07

Unfortunately the question is not very clear to me, so this answer is based on guessing and provided desired images. This means, that your MWE wasn't very helpful. From it I only see, that you expect figure drawn by pgfplots and not by pure tikz ...

See, if my result is what you after: Above images is done by the following code:

\documentclass[border=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{width=6cm, compat=1.13,
title style={at={(0.5,-0.1)}, anchor=north, font=\large},
xlabel style = {anchor=west},
ylabel style = {anchor=south},
clip=false
}% <-- common styles
\usetikzlibrary{arrows}

\begin{document}
\begin{tikzpicture}
\begin{axis}[%small,anchor=aninnernode.center,
axis y line=center,
axis x line=middle,
%
xmin=-0.5,xmax=5,
ymin=-0.5,ymax=5,
xtick={0}, ytick={0},% this disables the standard
ylabel=$y$,
xlabel=$x$,
title = {$f$ is increasing},
]
domain=0.5:4.5, samples=9, mark=none,
->] {0.2*(x^2) + 1};
\node[right] at (4.5,{0.2*(4.5^2) + 1})  {$f(x)$};
%
\draw[dashed]   (0,{0.2*(2^2) + 1}) node[left] {$f(x_1)$}  -|
(2, 0)              node[below] {$x_1$}
(0,{0.2*(3^2) + 1}) node[left] {$f(x_2)$}  -|
(3, 0)              node[below] {$x_2$};
\end{axis}
\end{tikzpicture}
\hfill
\begin{tikzpicture}
\begin{axis}[%small,anchor=aninnernode.center,
axis y line=center,
axis x line=middle,
%
xmin=-0.5,xmax=5,
ymin=-0.5,ymax=5,
xtick={0}, ytick={0},% this disables the standard
ylabel=$y$,
xlabel=$x$,
title = {$f$ is decreasing},
]
domain=0.5:4.5, samples=9, mark=none,
->] {7-3*sqrt(x)};
\node[right] at (4.5,{7-3*sqrt(4.5)})  {$f(x)$};
%
\draw[dashed]   (0,{7-3*sqrt(2)})   node[left] {$f(x_1)$}  -|
(2, 0)              node[below] {$x_1$}
(0,{7-3*sqrt(3)})   node[left] {$f(x_2)$}  -|
(3, 0)              node[below] {$x_2$};
\end{axis}
\end{tikzpicture}
\end{document}

The code is (almost) semi-explanatory, but if you have some question about it, pleas ask.

Addendum: The code for pure TikZ solution seems to be slightly simpler:

\documentclass[border=3mm]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[domain=0.5:4.5, samples=17,]
% axes
\draw[->] (-0.5,0) -- node[below=5mm] {$f$ is increasing}
+ (5.5,0) node[right] {$x$};
\draw[->] (0,-0.5) -- + (0,5.5) node[above] {$y$};
% graph
\draw[line width=2pt, draw=red, ->]
plot (\x,{0.2*(\x^2) + 1}) node[right] {$f(x)$};
%
\draw[dashed]   (0,{0.2*(2^2) + 1}) node[left]  {$f(x_1)$}  -|
(2, 0)              node[below] {$x_1$}
(0,{0.2*(3^2) + 1}) node[left]  {$f(x_2)$}  -|
(3, 0)              node[below] {$x_2$};
\end{tikzpicture}
\hfill
\begin{tikzpicture}[domain=0.5:4.5, samples=17]
% axes
\draw[->] (-0.5,0) -- node[below=5mm] {$f$ id decreasing}
+ (5.5,0) node[right] {$x$};
\draw[->] (0,-0.5) -- + (0,5.5) node[above] {$y$};
% graph
\draw[line width=2pt, draw=red, ->]
plot (\x,{7-3*sqrt(\x)}) node[right]  {$f(x)$};
%
\draw[dashed]   (0,{7-3*sqrt(2)})   node[left] {$f(x_1)$}  -|
(2, 0)              node[below] {$x_1$}
(0,{7-3*sqrt(3)})   node[left] {$f(x_2)$}  -|
(3, 0)              node[below] {$x_2$};
\end{tikzpicture}
\end{document}

Result of above MWE is (almost) the same as at the first solution (with use of pgfplots)

• actually morning :-) ...please, get me more information about my answer. provide it what you looking for? as i already noted, i have problems to understand what is your problem. For example, do you like to have pure TikZ solution, without use of the pgfplots? Jun 27 '16 at 2:26
• Actually, I would prefer pure tikzpicture to pgfplots but either am satisfied with your code and it looks good. Thank you. Jun 27 '16 at 2:36
• Now I added "pure TikZ" solution ... Jun 27 '16 at 3:49