How many functions f from the reals to the reals are there, such that f(f(x))=x^2−2?
To find the number of functions f from the reals to the reals that satisfy the equation f(f(x)) = x^2 - 2, we need to break down the problem step by step.
Step 1: Determine the potential values of f(x).
Let's consider the equation f(f(x)) = x^2 - 2. The right side of the equation, x^2 - 2, can take any real value. Therefore, the left side, f(f(x)), must also be able to take any real value. This implies that the range of f(x) should be all real numbers.
Step 2: Determine the possible values of f(y) for a given y.
From step 1, we know that f(y) can take any real value. Now, this means that for every y in the real numbers, there exists at least one x such that f(x) = y. So, for every real value y, there is always at least one x where f(x) = y.
Step 3: Count the number of possible functions.
Since for any real value y, there exists at least one x where f(x) = y, we can conclude that the number of functions f from the reals to the reals that satisfy f(f(x)) = x^2 - 2 is infinite.
Therefore, the answer to the question is that there are infinitely many functions f from the reals to the reals that satisfy the given equation.