Question

The domain of the function $ y = f(x) $, where $ x $ and $ y $ are related by $ 2^x + 2^y = 2 $, is:

• A. \( (-\infty, \infty) \)
• B. \( (-\infty, 1) \)
• C. \( (0, \infty) \)
• D. \( (1, \infty) \)

Hint:

For functional relationships involving exponentials like \( 2^x + 2^y = \text{constant} \), ensure the expressions on both sides are defined and positive.

Answer 1

The Correct Option is B

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: \[ (-\infty, 1) \]