Is it a Power Function or a Function in a Power?

Algebra Level 4

\[ \large 2^x + 2^{f(x)} = 2 \]

Consider a real valued function satisfying the equation above. Find the range of the function \(f(x) \).

Note: The domain of \( f(x) \) is the largest subset of \( \mathbb{R} \) allowed in which there exists a solution to the above equation.

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