# Solve for F: Find f(x+y)=f(x)(a-y) + f(y)f(a-x), f(0)=1/2

• Penultimate
In summary, the conversation discusses finding a function F from the real numbers to the real numbers that satisfies two given rules. The first rule involves the function being equal to the product of its inputs and a given constant, while the second rule sets the function's value at 0 to be 1/2. However, it is determined that such a function does not exist.
Penultimate
Find all the fumction F: R$$\rightarrow$$R So that :
1) f(x+y)=f(x)(a-y) + f(y)f(a-x)
2)f(0)=1/2

I dont know if i posted in the right place , but if not please moderators send me an email.
Thanks...

Penultimate said:
Find all the fumction F: R$$\rightarrow$$R So that :
1) f(x+y)=f(x)(a-y) + f(y)f(a-x)
2)f(0)=1/2

I dont know if i posted in the right place , but if not please moderators send me an email.
Thanks...

I suppose a is a given constant and first rule is for all x, y from R.

First assume x=y=0: 1/2 = 1/2 a + 1/2 f(a) $$\Rightarrow$$ f(a) = 1-a (1)
Then x=0 and y=a: f(a) = 1/2 . 0 + f(a)2 $$\Rightarrow$$ f(a) = 0 or 1 (2)
f(a) can't be 1, because then according (1) a=0 and f(0) is 1/2.
Let's assume x=a and y=0: f(a) = f(a)2 + 1/4 and f(a)=0 $$\Rightarrow$$ 0=1/4

This function can't exist.

## 1. What does "solve for F" mean in this equation?

In this equation, "solve for F" means to find the function f(x) that satisfies the given conditions.

## 2. What is the value of f(x) in the equation?

The value of f(x) is not specified in the equation, as it is a variable representing the function we are trying to find.

## 3. How do you find f(x) in this equation?

To find f(x), we need to manipulate the equation using algebraic operations and the given conditions to isolate f(x) on one side of the equation.

## 4. What does the condition f(0)=1/2 mean in this equation?

This condition means that when x=0, the value of f(x) is equal to 1/2. It is an initial condition that helps us in solving for f(x).

## 5. Can this equation be solved for multiple values of f(x)?

Yes, this equation can have multiple solutions for f(x). However, we can use the initial condition f(0)=1/2 to narrow down the possible solutions and find a unique function that satisfies all the given conditions.

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