# How do you find the domain and range of 2^x=y?

$x \in \mathbb{R} , y \ge 0$

#### Explanation:

The Domain is the list of all $x$ values that are allowable in the function.

When answer Domain questions, I like to look at what happens when I set $x = 0 , \infty , - \infty$ and any other value that might cause the function to not operate.

When $x = 0 , y = {2}^{0} = 1$ and so no problem there
When $x = \infty , y = {2}^{\infty} = \infty$ and so no problem there
When $x = - \infty , y = {2}^{- \infty} = \frac{1}{{2}^{\infty}} = 0$ and so no problem there

In fact, there are no $x$ values that are disallowed, and so we can say that all real numbers are in the list of allowable values. We can write that in math terms as:

$x \in \mathbb{R}$

The Range is the list of all resulting Domain values (and is usually the list of $y$ values). We've already seen that $y$ can be 0, can be 1, can be infinitely big. We can't make $y$ be less than 0 - there is no $x$ value we can plug in that will make $y$ negative. We can write that as:

$y \ge 0$

We can see this is the graph of $y = {2}^{x}$:

graph{2^x [-5,10,-5,20]}