Function can be tricky

Algebra Level 3

Let \(f(x)\) be a function with non-negative integer coefficient(s) but with power(s) of \(x\) not necessary integer(s) such that:

\(\begin{cases} f(0) & =2046 \\ f(1) & =2047 \\ f(2048) & =2048 \end{cases} \)

Find \(f(177147)\).

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