# How do you determine the domain, range and horizontal asymptote of each exponential function  f(x) = 1 - 2^[(-x/3)]?

Nov 18, 2017

See below.

#### Explanation:

There are no restrictions on $x$ so domain is:

$\left\{x \in \mathbb{R}\right\}$

For positive $x$ we have:

$y = 1 - \frac{1}{2} ^ \left(\frac{x}{3}\right)$

as $x \to \infty \textcolor{w h i t e}{8888} 1 - \frac{1}{2} ^ \left(\frac{x}{3}\right) \to 1 \textcolor{w h i t e}{88}$ ( horizontal asymptote )

For negative $x$ we have:

$y = 1 - {2}^{\frac{x}{3}}$

as $x \to - \infty \textcolor{w h i t e}{8888} 1 - {2}^{- \frac{x}{3}} \to - \infty$

So range is:

$\left\{y \in \mathbb{R} : - \infty < y < 1\right\}$

Graph: