Suppose A is a nonempty set and f: A ? A. Define f to be a constant function if

Question

Suppose A is a nonempty set and f: A ? A. Define f to be a constant function if there is some a ? A such that ? x ? A(f(x)=a).
Prove that f is a constant function if and only if for all g: A? A, f ? g = f.
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I think I have the first direction. Let x, y be arbitrary elements of A. And let (x,y) ? g. Suppose f is a constant function, then f ? g(x) = f(g(x)). Now since (x,y) ? g, g(x)=y, so we have f(g(x)) = f(y). And since y ? A, f(y)=a. so f? g=f.

I am unclear how to prove the other direction. Please verify the above and help me prove that f is a constant function if for all g: A? A, f ? g = f.
Thank you

Explanation / Answer

Say f is constant then f(g(x)) = f(x) for all x that is true. as for all x in A f(x) = a Then f(g(x)) = a = f(x) for all x Hence done. Part 2: given that for all g:A->A f(g(x))=f(x) To prove that f is constant, we will choose some proper g Choose g(x)=b for all x in A So f(g(x)) = f(b) = f(x) for all x So f(x) = f(b) for all x in A Hence f is constant. If you have any doubt, just message me.

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