How can I get the correct result with this picture?

Question

I want to draw like this picture

I tried.

First way (I got the correct result)

\documentclass[12pt,border=1mm]{standalone} 
\usepackage{fouriernc} 
\usepackage{pgfplots} 
\pgfplotsset{compat=1.16} 
\begin{document} 
\begin{tikzpicture}[>=stealth,declare function={ 
xmin=-1;xmax=10;ymin=-3;ymax=2;}] 
\draw[gray!30] (xmin,ymin) grid (xmax,ymax); % grid 
\draw[->, thick] (xmin,0)--(xmax,0) node [below left]{$x$}; 
\draw[->,thick] (0,ymin)--(0,ymax) node [below left]{$y$}; 
\node[below right] at (0, 0) {$O$}; 
\foreach \p/\g in {1/-90,2/-45,3/-90,4/90,5/-90,6/-45,7/-90,8/-90,9/-90 }\draw(\p,0)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\foreach \p/\g in {-2/180,1/180}\draw(0,\p)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\draw[very thick] (2,0)
arc [start angle=0, end angle=180, radius=1];
\draw[very thick] (6,0)
arc [start angle=0, end angle=-180, radius=2];
\draw[thick] (6,0) -- (9,1);
\foreach \Y in {(2,0),(4,-2),(6,0),(9,1)} \fill \Y circle(2pt); 
\draw[dashed] (9,0) |- (0,1);
\draw[dashed] (4,0) |- (0,-2);
\end{tikzpicture} 
\end{document}

Second way (I didn't get the correct result)

\documentclass[12pt,border=1mm]{standalone} 
\usepackage{fouriernc} 
\usepackage{pgfplots} 
\pgfplotsset{compat=1.16}
\begin{document} 
\begin{tikzpicture}[>=stealth,declare function={ 
xmin=-1;xmax=10;ymin=-3;ymax=2;
f(\x)=sqrt(2*\x- \x*\x);
g(\x)=-sqrt(-12 + 8*\x - \x*\x);
}] 
%\draw[gray!30] (xmin,ymin) grid (xmax,ymax); % grid 
\draw[->, thick] (xmin,0)--(xmax,0) node [below left]{$x$}; 
\draw[->,thick] (0,ymin)--(0,ymax) node [below left]{$y$}; 
\node[below right] at (0, 0) {$O$}; 
\foreach \p/\g in {1/-90,2/-45,3/-90,4/90,5/-90,6/-45,7/-90,8/-90,9/-90 }\draw(\p,0)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\foreach \p/\g in {-2/180,1/180}\draw(0,\p)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\draw[thick] (6,0) -- (9,1);
%\foreach \Y in {(2,0),(4,-2),(6,0),(9,1)} \fill \Y circle(2pt); 
\draw[dashed] (9,0) |- (0,1);
\draw[dashed] (4,0) |- (0,-2);
\draw[very thick,blue,smooth,samples=250,domain=0:2] plot(\x,{f(\x)});
\draw[very thick,blue,smooth,samples=250,domain=2:6] plot(\x,{g(\x)});  
\end{tikzpicture} 
\end{document}

How can I get the correct result with second way?

Answer 1

According to this awesome answer you need to put plot function instead of only add plot.

I have tried and I could not find a way to use plot function within f(\x) or g(\x), so I deleted them and replaced by their definitions that were in declare function to get your first way (btw I do not know why you forget to add blue color, the black points and grid background):

\documentclass[12pt,border=1mm]{standalone} 
\usepackage{fouriernc} 
\usepackage{pgfplots} 
\pgfplotsset{compat=1.15}
\begin{document} 
\begin{tikzpicture}[>=stealth,declare function={ 
xmin=-1;xmax=10;ymin=-3;ymax=2;
%f(\x)=sqrt(2*\x- \x*\x); % deleted
%g(\x)=-sqrt(-12 + 8*\x - \x*\x); % deleted
}] 
%\draw[gray!30] (xmin,ymin) grid (xmax,ymax); % grid 
\draw[->, thick] (xmin,0)--(xmax,0) node [below left]{$x$}; 
\draw[->,thick] (0,ymin)--(0,ymax) node [below left]{$y$}; 
\node[below right] at (0, 0) {$O$}; 
\foreach \p/\g in {1/-90,2/-45,3/-90,4/90,5/-90,6/-45,7/-90,8/-90,9/-90 }\draw(\p,0)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\foreach \p/\g in {-2/180,1/180}\draw(0,\p)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\draw[thick] (6,0) -- (9,1);
%\foreach \Y in {(2,0),(4,-2),(6,0),(9,1)} \fill \Y circle(2pt); 
\draw[dashed] (9,0) |- (0,1);
\draw[dashed] (4,0) |- (0,-2);
\draw[very thick,blue,smooth,samples=250,domain=0:2] plot function{sqrt(2*x- x*x)};
\draw[very thick,blue,smooth,samples=250,domain=2:6] plot function{-sqrt(-12 + 8*x - x*x)};  
\end{tikzpicture} 
\end{document}

Answer 2

I do not know why this happens (seems that this has to do with the conversion between cm and pt and some numerical errors) and manoooh's answer is the cleaner way to go. This is just in case you do not have gnuplot installed. Then you can just "continue" the functions in such a way that they remain well-defined and plot over a slightly longer interval.

\documentclass[12pt,border=1mm]{standalone} 
\usepackage{fouriernc} 
\usepackage{pgfplots} 
\pgfplotsset{compat=1.16}
\begin{document} 
\begin{tikzpicture}[>=stealth,declare function={ 
xmin=-1;xmax=10;ymin=-3;ymax=2;
f(\x)=ifthenelse(2*\x- \x*\x>0,sqrt(abs(2*\x- \x*\x)),0);
g(\x)=-ifthenelse(-12 + 8*\x - \x*\x>0,sqrt(abs(-12 + 8*\x - \x*\x)),0);
}] 
%\draw[gray!30] (xmin,ymin) grid (xmax,ymax); % grid 
\draw[->, thick] (xmin,0)--(xmax,0) node [below left]{$x$}; 
\draw[->,thick] (0,ymin)--(0,ymax) node [below left]{$y$}; 
\node[below right] at (0, 0) {$O$}; 
\foreach \p/\g in {1/-90,2/-45,3/-90,4/90,5/-90,6/-45,7/-90,8/-90,9/-90 }\draw(\p,0)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\foreach \p/\g in {-2/180,1/180}\draw(0,\p)node[shift={(\g:.3)},scale=1]{$\p$}--+(0,.05)--+(0,-.05); 
\draw[thick] (6,0) -- (9,1);
%\foreach \Y in {(2,0),(4,-2),(6,0),(9,1)} \fill \Y circle(2pt); 
\draw[dashed] (9,0) |- (0,1);
\draw[dashed] (4,0) |- (0,-2);
\draw[very thick,blue] plot[smooth,samples=250,domain=0:2.003](\x,{f(\x)});
\draw[very thick,blue] plot[smooth,samples=250,domain=2:6.003](\x,{g(\x)});  
\end{tikzpicture} 
\end{document}