Suppose that f:R→R is a continuous function and satisfies the equation f(x)f(f(x))=1 for all x∈R. Further, if f(1000)=999, find the range of the function f(x).
Let y be a value in the range of f(x). Then y*f(y) = 1, which implies f(y)=1/y. Then f(x) = 1/x for any x falling in the range of f(x).
Let z be a value such that 1/z is in the range of f(x). Then (1/z)*f(1/z)=1, which implies f(1/z)=z. Thus, if x falls in the range of f(x) then 1/x will also fall in the range. Notice that this also means if x is not in the range then 1/x is also not in the range.
f(1000)=999 then means 999 is in the range and 1000 is not. Then the range of f(x) can be expressed as an interval [1/k, k] such that 999<=k<1000.
Solution
