Question

The domain of the function f(x)
${2}^{x} + {2}^{y} = 2$

Answer 1

HELLO THERE!

What actually is meant by Domain?

Domain means, the set of values of the independent variable, such that the function is real. For example, if a function is:

$f(x) = \frac{1}{x}$,

then the domain of the function (value of x) is R - {0} (i.e., all real numbers except 0). This is because, when x = 0, the function is undefined.

Now let's go to your question:

$2^{x} + 2^{y} = 2$

Here, the dependent variable and the independent variables are not clearly noted, so we must find the domain of x by taking x as an independent variable and y as dependent variable, and domain of y by taking y as an independent variable and x as an independent variable. For this, you must know the concept of Logarithm.

For domain of x, let’s assume y as a function of x:

$y = log_{2}(2 - 2^{x})$

Hence, it is clear that:

$2 - 2^{x} > 0$

$\implies 2^{x} < 2$

or x < 1.

So, for domain of x, we have: x ϵ (−∞, 1)

For Domain of y let’s assume x as a function of y:

$x = log_{2}(2 - 2^{y})$

Hence, it is clear that:

$2 - 2^{y} > 0$

$\implies 2^{y} < 2$

or, y < 1.

So, for domain of y, we have: y ϵ (−∞, 1)

So, your answers are:

x ϵ (−∞, 1)

y ϵ (−∞, 1)

THANKS!