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

May 9, 2017

The domain is $\left(- \infty , \infty\right)$ and the range is $\left(0 , \infty\right)$.

#### Explanation:

Given:

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

First note that this is well defined for any real value of $x$.

So the domain is the whole of $\mathbb{R}$, or in interval notation $\left(- \infty , \infty\right)$.

Next note that ${2}^{- x} > 0$ for any real value of $x$. So the range is $\left(0 , \infty\right)$ or a subset of it.

Let $y > 0$. Then $\log \left(y\right)$ is well defined. So we can take logs of both sides to find:

$\log \left(y\right) = \log \left({2}^{- x}\right) = \left(- x\right) \log \left(2\right)$

Dividing both sides by $- \log \left(2\right)$ we find:

$x = - \log \frac{y}{\log} \left(2\right)$

So that tells us that for any $y > 0$ there is an $x$ such that:

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

So the range is the whole of $\left(0 , \infty\right)$.