Animated Solution for Mathematics - Functions: The domain of definition of the function given by the equation $2^x+2^y=2$ is

Welcome, fellow explorer of the mathematical landscape. Today, we are going to dissect a problem that seems deceptively simple but hides a profound truth about the nature of exponential functions.

We are tasked with finding the domain of the equation $2^x+2^y=2$. At first glance, it looks like a standard algebraic equation, but the moment we ask, "For which values of $x$ does this equation hold?", we enter the realm of domain analysis.

To understand the domain, we must isolate the variable $y$. The domain of $x$ is defined by the values for which a corresponding real $y$ exists.

Let us rearrange our given equation, $2^x+2^y=2$, to isolate the $y$-term. By subtracting $2^x$ from both sides, we obtain the expression:

This is our raw setup. We have successfully expressed the exponential term $2^y$ entirely in terms of $x$. Now, the question becomes: what constraints must $x$ satisfy for this equation to yield a valid real $y$?

Here is where the magic happens. Think about the left-hand side: $2^y$.

In the world of real numbers, can an exponential function with a positive base ever be zero or negative? Never! For any real number $y$, $2^y$ is strictly greater than zero.

This is a fundamental, non-negotiable property of the exponential function. If the left-hand side is strictly positive, then the right-hand side must also be strictly positive for the equality to hold. This gives us our crucial inequality:

Now, we are on familiar ground. We need to solve $2-2^x>0$.

Moving $2^x$ to the right side, we get $2>2^x$, or more intuitively:

To solve this, we can write $2$ as $2^1$. So, the inequality becomes $2^x<2^1$.

Since the base $2$ is greater than $1$, the exponential function $f(x)=2^x$ is an increasing function. This means that if $2^x<2^1$, then the exponents must satisfy the same inequality:

We have arrived at our destination. The condition for the existence of $y$ is simply $x<1$.

Graphically, this means that our function is defined only for values of $x$ to the left of the vertical line $x=1$. In interval notation, the domain is $(-\infty ,1)$.

It is a beautiful result, isn't it? By simply respecting the inherent constraints of the exponential function, we have unlocked the domain of this equation. Keep this logic in your toolkit—whenever you see an exponential term, always check its range. It is often the key to the entire problem.