Solving Functional Equations: Tips & Examples

[stanley.st]

Hello,

could you explain me what's the right way to solve these equations. I've never solved it before.

$$f(x+y)+f(x-y)=2f(x)f(y)\,\;\;\forall x,y\in\mathbb{R}$$
$$f(x)+\left(x+\frac{1}{2}\right)f(1-x)=1\,\;\;\forall x\in\mathbb{R}$$

thank you...

 

[slider142]

Hello,

could you explain me what's the right way to solve these equations. I've never solved it before.

$$f(x+y)+f(x-y)=2f(x)f(y)\,\;\;\forall x,y\in\mathbb{R}$$
$$f(x)+\left(x+\frac{1}{2}\right)f(1-x)=1\,\;\;\forall x\in\mathbb{R}$$

thank you...

Since the equation is true for all values, use clever values of x and y to get a system of equations.

 

[HallsofIvy]

For example, if, in the first, we take y= 0, we get f(x+0)+ f(x- 0)= 2f(x)= 2f(x)f(0) so we must have f(0)= 1. But if we take x= 0, we get f(0+y)+ f(0- y)= f(y)+ f(-y)= 2f(0)f(y) = 2f(y). That is, f(-y)= f(y) for all y so f is an even function.