We have a function \(g : \mathbb{R} \to \mathbb{R}\) that satisfies \(g(x)g(y) = g(x + y) + xy\)
for some real numbers \(x\) and \(y\). Find all possible functions \(g\).
Please help! I think a good place to start is plugging in combinations of 1 and 0, but what would I do after plugging in all the combinations?
1) Try x = 0.
2) Try x = y
3) Try y = 0
4) Try x = 1
5) Try y = 1
6) Try f(x) = c
Conclusion: There are no functions that work
+33661 g(x) = x + 1;
g(y) = y + 1
g(x + y) = x + y + 1
g(x)*g(y) = x*y +x + y + 1
g(x+y) + x*y = x + y + 1 + x*y
Thanks so much for the response. I understand everything past the first step, but I'm not exactly sure how you got there. How did you know that g(x)=x+1 and that g(y)=y+1? Thanks again.
+33661 You are trying to find functions that satisfy the relationship: g(x)*g(y) = g(x+y) + x*y
Pretend I defined the function as g(z) = z + 1.
When z = x we have g(x) = x+1
When z = y we have g(y) = y + 1
When z = x+y we have g(x+y) = x + y + 1
So this definition of the function g satisfies the specified criterion.
I suspect its the only one, but I haven't attempted to prove it!
