The domain of the function $ y = f(x) $, where $ x $ and $ y $ are related by $ 2^x + 2^y = 2 $, is:
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For functional relationships involving exponentials like \( 2^x + 2^y = \text{constant} \), ensure the expressions on both sides are defined and positive.
Step 1: Given the relation between \( x \) and \( y \):
\[
2^x + 2^y = 2.
\]
Step 2: Rearranging to isolate \( 2^y \):
\[
2^y = 2 - 2^x.
\]
Since \( 2^y>0 \) for all real \( y \), we must have:
\[
2 - 2^x>0 \Rightarrow 2^x<2.
\]
Taking \(\log_2\) on both sides:
\[
x<1.
\]
Step 3: Domain of \( f(x) \):
The function \( y = f(x) \) exists only when \( x<1 \). Therefore, the domain is:
\[
(-\infty, 1)
\]