posted by Suzy on .
For 0 < x < 1, let
f(x) = (1 + x)(1 + x4)(1 + x16)(1 + x64)(1 + x256) · · ·
Compute f to the power of -1 times 8 divided by 5 times f times 3/8
First, we have to establish that the function f(x) is invertible on the interval [0,1].
f(x) consists of a polynomial with all positive terms, so it is strictly increasing and consequently one-to-one and onto. Thus f-1(x) exists.
Where x can be solved explicitly for y, an analytic expression of the inverse can be found. In other cases, we can resort to numerical solutions, which can be obtained to any accuracy we wish. For the given problem, we will supply a numerical solution.
We start with a property of f-1(x). which can be looked at as
f-1(f(x)) = x for all x on the given interval.
Thus f-1(f(3/8))= 3/8.
To find y=f-1(8/5f(3/8)), we need to find y for which
We can find the approximation by the secant method.
Knowing f(0)=1, f(1)=32,
we give a first approximation of
The new y could be interpolated:
We get successively
but the answer above needs to be put in simpler words to understand it