Find f(x): SolutionSolve Functional Equation: f(x)

[spacetime]

Find $$f(x)$$

if

$$f(x) + f(\frac{x-1}{x}) = 1 + x$$

 

[stunner5000pt]

are you completely sure that the right hand side is x +1 and not x -1??

 

[wais]

To solve this functional equation, we can first substitute x = 1 into the equation to get:

f(1) + f(0) = 1 + 1

Since f(0) is not defined, we can assume it to be a constant k. Therefore, we have:

f(1) + k = 2

Solving for f(1), we get:

f(1) = 2 - k

Next, we can substitute x = 2 into the equation to get:

f(2) + f(\frac{1}{2}) = 1 + 2

Substituting f(1) = 2 - k, we get:

f(2) + f(\frac{1}{2}) = 3 - k

Solving for f(2), we get:

f(2) = 3 - k - f(\frac{1}{2})

Continuing this process, we can find the values of f(3), f(4), and so on. In general, we can see that f(x) is dependent on the values of f(\frac{x-1}{x}). Therefore, we can define f(x) recursively as:

f(x) = 1 + x - f(\frac{x-1}{x})

This function will satisfy the given functional equation and can be used to find the value of f(x) for any given x.