First slide

Functions (XII)

Question

The domain of definition of the function y (x) given by the equation 2^x + 2^y = 2 is

Moderate

A

0 < x ≤ 1
B

0 ≤ x ≤ 1
C

– ∞ < x ≤ 0
D

– ∞ < x < 1

Solution

$\begin{array}{l} We have, 2^{x} + 2^{y} = 2 \Rightarrow 2^{y} = 2 - 2^{x} \\ \Rightarrow y = \frac{\log (2 - 2^{x})}{\log 2} \end{array}$
For y to be defined, $2 - 2^{x} > 0 \Rightarrow 2^{x} < 2$
Since, 2^x is an increasing function, therefore we get x < 1.

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