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

Apr 27, 2018

The domain is $x \in \mathbb{R}$. The range is $y \in \left(0 , + \infty\right)$

#### Explanation:

The function is

$y = {2}^{- x} = \frac{1}{2} ^ x$

${\lim}_{x \to - \infty} y = {\lim}_{x \to - \infty} \frac{1}{2} ^ x = \infty$

${\lim}_{x \to + \infty} y = {\lim}_{x \to + \infty} \frac{1}{2} ^ x = 0$

$\forall x \in \mathbb{R} , \frac{1}{2} ^ x > 0$

The domain of $y$ is $x \in \mathbb{R}$

${2}^{x} = \frac{1}{y}$

$\ln \left({2}^{x}\right) = \ln \left(\frac{1}{y}\right)$

$x \ln 2 = \ln \left(\frac{1}{y}\right) = \ln 1 - \ln y = - \ln y$

$x = - \frac{1}{\ln} 2 \ln y$

Therefore,

$y \in \left(0 , + \infty\right)$

The range is $y \in \left(0 , + \infty\right)$

graph{2^(-x) [-10.81, 14.5, -2.89, 9.77]}