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?

  • $\begingroup$ 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

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 $g$.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.