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I have a question about filling out incomplete function tables. I am given the following set of function values and asked to fill in the missing ones:

Table of values of ffunctions f, g and h

I'm looking for a few hints as to how to approach this problem. I can see that I am able to trace a path from $f(0)$ to $g(2)$ to $h(3)$ for example but I'm not sure how to go about deriving the expressions of the functions.

Could somebody help me out?

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  • $\begingroup$ One example: $h(3) = g(f(3)) = 1$. But $f(3) = 0$, so we must have $g(f(3)) = g(0) = 1$. $\endgroup$ – rogerl Jun 17 '14 at 14:01
  • $\begingroup$ Thanks that helped a lot, I solved it now! $\endgroup$ – Irresponsible Newb Jun 17 '14 at 14:43
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As rogerl suggests in the comments, $g(0)=1$ because $0=f(3)$, and so $g(0)=g(f(3))=h(3)=1$.

  • Similarly, $g(3)=g(f(4))=h(4)=4$.
  • A more straightforward calculation: $h(0)=g(f(0))=g(2)=3$.

Strictly speaking, there is not enough information to determine $f(1)$ and $f(2)$; for instance if $f(1)=999$, and $g(999)=0$, then this makes $h(1)=0$ true and doesn't cause any other inconsistencies.

However, it's reasonable to assume that in this problem, $f,g$ and $h$ are each functions whose domains and codomains are both $\{0,1,2,3,4\}$. Under this assumption:

  • Note that $0=h(1)=g(f(1))$, and the only $y$ such that $g(y)=0$ is $y=1$. Therefore, $f(1)=1$.

  • Note that $2=h(2)=g(f(2))$, and the only $y$ such that $g(y)=2$ is $y=4$. Therefore, $f(2)=4$.

The reason these arguments fail without the assumption is because of the words the only. In particular, the only part of the assumption we –really– need, is that $\{0,1,2,3,4\}$ is the domain of $g$.

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  • $\begingroup$ This answer exists to remove this question from the Unanswered queue. Please upvote or accept this answer to complete the process. $\endgroup$ – aleph_two Oct 21 '18 at 5:17

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