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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?

 Jul 30, 2020
 #1
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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

 Jul 30, 2020
 #2
avatar+30931 
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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

 Jul 30, 2020
edited by Alan  Jul 30, 2020
 #3
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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.

Guest Jul 30, 2020
 #4
avatar+30931 
+1

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!

Alan  Jul 30, 2020
edited by Alan  Jul 30, 2020
 #5
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Ohh, ok. Thank you so much!

Guest Jul 30, 2020

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