# Filling out tables of incomplete functions

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:

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?

• One example: $h(3) = g(f(3)) = 1$. But $f(3) = 0$, so we must have $g(f(3)) = g(0) = 1$. – rogerl Jun 17 '14 at 14:01
• Thanks that helped a lot, I solved it now! – 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 $$g$$.

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