How many functions f from the reals to the reals are there, such that f(f(x))=x^2−2?

Details and assumptions
f need not be a polynomial. f need not even be continuous.

To determine the number of functions f from the real numbers to the real numbers such that f(f(x)) = x^2 - 2, we can approach this problem step by step.

Step 1: Analyze the equation f(f(x)) = x^2 - 2
The equation f(f(x)) = x^2 - 2 represents a functional equation, where we are looking for functions f that satisfy this equation for all real numbers x.

Step 2: Understand the properties of the equation
From the equation, we can infer that f(x) must be a function that maps the real numbers to themselves. Additionally, for any x, f(f(x)) must be equal to x^2 - 2.

Step 3: Consider function properties and transformations
Let's consider the properties of the function f. Since f(f(x)) = x^2 - 2, applying f to both sides of the equation gives f(f(f(x))) = f(x^2 - 2). However, we know that f(f(x)) = x^2 - 2, so we can substitute x^2 - 2 for f(f(x)), giving f(x^2 - 2) = x^2 - 2.

Step 4: Determine the possible solutions
The equation f(x^2 - 2) = x^2 - 2 can have multiple solutions, and there are infinitely many functions that can satisfy this equation. Therefore, there is not a unique solution to this functional equation.

Step 5: Conclusion
In summary, there are infinitely many functions f from the real numbers to the real numbers that satisfy the equation f(f(x)) = x^2 - 2. The existence of these functions is due to the flexibility and openness of the functional equation, which allows for a broad range of solutions.