The domain of definition of the function y(x) is given by… — JEE Main 2000 Mathematics PYQ

2000

The domain of definition of the function

y(x)

is given by the equation

2^x + 2^y = 2

, is

A.
$0 < x \leq 1$
B.
$0 \leq x \leq 1$
C.
$-\infty < x \leq 0$
D.
$-\infty < x < 1$

Show answer and solution

Correct option: D

Solution

From 2^x + 2^y = 2, we get 2^y = 2 - 2^x, so y = log₂(2 - 2^x).

For y to be defined, we need 2 - 2^x > 0, i.e., 2^x < 2 = 2¹, which gives x < 1. Since 2^x > 0 for all real x, there's no lower bound restriction, so x can be any value from -∞ up to (but not including) 1. Hence the domain is -∞ < x < 1.

[2000]

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