You could use PGFplots for this. By plotting functions using
\addplot [restrict y to domain=0.5:inf] {(x-2)>0};
all coordinates that do not fulfill the condition will be discarded.
I've defined an environment functionallines of the form
\begin{functionallines}[<pgfplots arguments>]{<zero positions>}{<lower>:<upper>}
The <zero positions> argument takes a comma-separated list of x-values where any of the functions are zero or not defined. At these positions, a vertical line will be drawn through the function plots, and the positions will be printed in black at the top of the diagram. If you want to change the labels for these positions, use the <pgfplots arguments> to supply extra x tick labels={<first label>,<second label>,...}.
The optional argument can also be used for things like switching off the regular tick labels by setting xtick=\empty, or changing the drawing style of the functions by, for example, setting negative/.style={orange,dashed}, positive/.style={green, ultra thick}.
The <lower> and <upper> arguments define the x-range of the plot.
You can then add new lines using
\functionalline[<LaTeX math code>]{<expression>}{<vertical position>}{<list of zeros>}
where the optional argument [<LaTeX math code>] can be used to specify the code used to print the equation.
If the function evaluates to a nonreal value (divide by zero) at one of the zero positions, an x is printed instead of a 0.
Here are a couple of examples:
\begin{functionallines}{-1,1}{-2:2}
\functionalline{x-1}{1}{1}
\functionalline{x+1}{2}{-1}
\functionalline[f(x) = \dfrac{(x-1)}{(x+1)}]{(x-1)/(x+1)}{3.5}{-1,1}
\end{functionallines}
\begin{functionallines}[
xtick=\empty,
negative/.style={orange,dashed},
positive/.style={green, ultra thick}
]{0,90,180,270,360}{-20:380}
\functionalline[\cos(x)]{cos(x)}{1}{90,270}
\functionalline[\sin(x)]{sin(x)}{2}{0,180,360}
\functionalline[\cos(x) \cdot \sin(x)]{cos(x) * sin(x)}{3}{0,90,180,270,360}
\end{functionallines}
Setting the labels for precise zero positions using extra x ticks labels, specifying the normal ticks using xtick={<list>} to avoid overlaps.
\begin{functionallines}[
extra x tick labels={$\sqrt{5}$,$\pi$,$2\pi$},
xtick={0,1,4,5}]{2.23,3.14,6.3}{0:6.5}
\functionalline[x-\sqrt{5}]{x-2.23}{1}{2.23}
\functionalline[\sin(x)]{sin(x*180/3.14)}{2}{3.14,6.28}
\functionalline[(x-\sqrt{5})\cdot \sin(x)]{(x-2.23)*sin(x*180/3.14)}{3}{2.23,3.14,6.28}
\end{functionallines}
And here's the complete code:
\documentclass{article}
\usepackage{pgfplots}
\usepackage{amsmath}
\begin{document}
\pgfplotsset{
shift down/.style={
y filter/.code={\pgfmathparse{\pgfmathresult*(#1)}}
},
shift down/.default=1,
every axis plot post/.style={restrict y to domain=0.5:inf},
positive/.style={
no markers,
red
},
negative/.style={
no markers,
blue
},
/tikz/function label/.style={
anchor=east
},
step functionallinenumber/.code={
\stepcounter{functionallinenumber}
},
title entries/.initial={}
}
\makeatletter
\newcommand\functionalline[4][\@empty]{
\edef\plots{
\noexpand\addplot [negative, shift down=#3, forget plot] {#2<0};
\noexpand\addplot [positive, shift down=#3, forget plot] {#2>0};
}
\plots
\node at (axis cs:\pgfkeysvalueof{/pgfplots/xmin},#3) [function label] {%
\ifx#1\@empty%
$#2$%
\else%
$#1$%
\fi
};
\pgfplotsinvokeforeach {#4} {
\node at (axis cs:##1,#3) [
fill=white,
inner sep=1pt,
declare function={x=##1;} % Set 'x' to current position
] {%
\pgfkeys{/pgf/fpu}% Use the fpu library, because it doesn't throw an error for divide by zero, but sets result to +/- inf
\pgfmathparse{#2}%
\pgfmathfloatifflags{\pgfmathresult}{0}{\hspace{-0.75ex}0}{x}% Check whether result is zero. The \hspace is necessary because of a bug in the fpu library. (Update 11 June 2012: Doesn't seem to be the case anymore, the \hspace can be removed)
\pgfkeys{/pgf/fpu=false}%
};
}
}
\newenvironment{functionallines}[3][]{
\begin{tikzpicture}
\begin{axis}[
extra x ticks = {#2},
grid=none,
xticklabel pos=right,
hide y axis,
x axis line style={draw=none},
every tick label/.style={
anchor=base,
yshift=1ex,
gray!50
},
every extra x tick/.style={
every tick label/.style={
anchor=base,
yshift=1ex,
inner xsep=0pt,
fill=white,
text=black
}
},
extra x tick style={grid=major},
xtick pos=right,
major tick length=0pt,
enlarge x limits=false,
enlarge y limits={abs=0.75},
domain=#3,
samples=100,
y dir=reverse, y = -0.5cm,
clip=false,
#1
]
}{
\coordinate (bottom right) at (rel axis cs:1,0);
\coordinate (top right) at (rel axis cs:1,1);
\end{axis}
\draw [-latex] (top right-|current bounding box.west) -- (top right) node [right] {$x$};
\draw (bottom right) -- (bottom right-|current bounding box.west);
\end{tikzpicture}
}
\begin{functionallines}{-1,1}{-2:2}
\functionalline{x-1}{1}{1}
\functionalline{x+1}{2}{-1}
\functionalline[f(x) = \dfrac{(x-1)}{(x+1)}]{(x-1)/(x+1)}{3.5}{-1,1}
\end{functionallines}
\hspace{1cm}
\begin{functionallines}[xtick=\empty]{0,90,180,270,360}{-20:380}
\functionalline[\cos(x)]{cos(x)}{1}{90,270}
\functionalline[\sin(x)]{sin(x)}{2}{0,180,360}
\functionalline[\cos(x) \cdot \sin(x)]{cos(x) * sin(x)}{3}{0,90,180,270,360}
\end{functionallines}
\hspace{1cm}
\begin{functionallines}[
extra x tick labels={$\sqrt{5}$,$\pi$,$2\pi$},
xtick={0,1,4,5}]{2.23,3.14,6.3}{0:6.5}
\functionalline[x-\sqrt{5}]{x-2.23}{1}{2.23}
\functionalline[\sin(x)]{sin(x*180/3.14)}{2}{3.14,6.28}
\functionalline[(x-\sqrt{5})\cdot \sin(x)]{(x-2.23)*sin(x*180/3.14)}{3}{2.23,3.14,6.28}
\end{functionallines}
\end{document}
xabove the equations. Also, am I right in assuming that the first and second equation are swapped in your example pictures?