# How do you graph f(x)=-(2/3)^x+3 and state the domain and range?

Nov 11, 2017

See below.

#### Explanation:

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

First find any $y$ axis intercepts. These will occur where $x = 0$:

$y = - {\left(\frac{2}{3}\right)}^{0} + 3 = 2$

$y$ axis intercept at $\left(0 , 2\right)$

$x$ axis intercept when $y = 0$

$- {\left(\frac{2}{3}\right)}^{x} + 3 = 0$

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

$x \ln \left(\frac{2}{3}\right) = \ln \left(3\right) \implies x = \left(\ln \frac{3}{\ln \left(\frac{2}{3}\right)}\right) \approx - 2.71$

$\left(- 2.71 , 0\right)$

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

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

The line $y = 3$ is a horizontal asymptote:

Graph: