定义用于制作功能线/包的新命令

Question 1

你可以使用 PGFplots 来实现这一点。通过使用

 \addplot [restrict y to domain=0.5:inf] {(x-2)>0};

所有不满足条件的坐标将被丢弃。

functionallines我定义了以下形式的环境

\begin{functionallines}[<pgfplots arguments>]{<zero positions>}{<lower>:<upper>}

该<zero positions>参数采用逗号分隔的 x 值列表，其中任何函数为零或未定义。在这些位置，将绘制一条穿过函数图的垂直线，并且这些位置将以黑色打印在图表顶部。如果您想更改这些位置的标签，请使用来<pgfplots arguments>提供extra x tick labels={<first label>,<second label>,...}。

可选参数还可用于诸如通过设置来关闭常规刻度标签xtick=\empty，或通过设置来更改函数的绘制样式等negative/.style={orange,dashed}, positive/.style={green, ultra thick}。

<lower>和参数<upper>定义了图的 x 范围。

然后可以使用

\functionalline[<LaTeX math code>]{<expression>}{<vertical position>}{<list of zeros>}

其中可选参数[<LaTeX math code>]可用于指定用于打印方程的代码。

如果函数在其中一个零位置处计算出非实数值（除以零），则会x打印而不是0。

以下是几个示例：

\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}

使用设置精确零位置的标签extra x ticks labels，使用指定正常刻度xtick={<list>}以避免重叠。

\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}

完整代码如下：

\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}

Answer