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I just started learning Functional Equations and I was working on a problem that asked me to find all functions satisfying a certain condition, and at some point I got $f(x)=xf(1)$, is there a way to express $f(x)$ only in terms of $x$?, or is that an answer already?

Thank you.

Mualpha7
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    If this is the only constraint, then any function $f(x)=cx$ will work (setting $c=f(1)$). You are already done. – Mario Carneiro Apr 03 '14 at 00:37
  • Oh, OK, Thanks! But I want to know: Can I find the numerical value of that constant? – Mualpha7 Apr 03 '14 at 00:41
  • Ah, I understand. The problem asked me to find all the functions that satisfy a given condition, and $f(x)=cx$ is already an answer because although we do not know the values that $f(x)=cx$ is going to output for each $x$ (because we do not know what $c$ is), we have that $f(x)=cx$ is well-defined. – Mualpha7 Apr 03 '14 at 00:56

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If by answer you mean $f(x)$ in terms of $x$ only, then $f(x)=xf(1)$ is an answer already iff $f(1)$ does not have any $x$ term in it.

  • But can I get a numerical answer for $f(1)$? – Mualpha7 Apr 03 '14 at 00:39
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    @Mualpha7 I don't know if that's possible given just that much information. Substituting $x=1$ into $f(x)=xf(1)$ gives the trivial equality, $f(1)=f(1)$. Thus, you need more information for a numerical answer for $f(1) $. –  Apr 03 '14 at 00:42