# How to solve and draw the graph of this function equation

I see a graph of a function equation in the title page of this book, but the specific drawing method is not given in the book. I want to know how to solve this function equation and draw its image:

$$f(x)+f(2x)+f(3x)=0$$

• Could you write what is $f(x)$ ? – Claude Leibovici Jul 29 '20 at 5:45
• @ClaudeLeibovici This is an equation for the function f, similar to this example: WolframAlpha["solve f(f(x))=x", {{"SolutionAsAFunctionalEquation", 1}, "Content"}]. But I can't find one of its concrete expressions at present. – Please correct GrammarMistakes Jul 29 '20 at 6:40
• Interesting question. By eye and ruler some of the zeros appear to be approximately $0.6, 1.2,1.8$. Also $0.4, 0.8,1.2$. $f(0) = 0$ – arthur Jul 29 '20 at 6:48
• Also $0.2,0.4,0.6$ – arthur Jul 29 '20 at 7:02
• I guess they ( "the book people" ) chose an even function with some definition in an interval. Other values are found with the recurrence. – Felix Marin Aug 31 '20 at 16:19

$$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$$ I guess they ( "the book people" ) chose
• Other values are found with the recurrence. \begin{align} \mbox{Namely,}\quad\mrm{f}\pars{x} & = \left\{\begin{array}{lcl} \ds{\phantom{-}\mrm{f}\pars{-x}} & \mbox{if} & \ds{x < 0} \\[2mm] \ds{\phantom{-}x\sin\pars{63x^{1/7}}} & \mbox{if} & \ds{0 \leq x \leq 1} \\[2mm] \ds{-\,\mrm{f}\pars{x \over 3} - \mrm{f}\pars{2x \over 3}}&&\mbox{otherwise} \end{array}\right. \end{align} $$\ds{\large\underline{\mbox{The Result}}}:$$