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

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Answer 1 · 2017-11-11T12:06:45

See below.

$y=-(2/3)^x+3$

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

$y=-(2/3)^0+3=2$

$y$ axis intercept at $( 0 , 2 )$

$x$ axis intercept when $y=0$

$-(2/3)^x+3=0$

$(2/3)^x=3$

$xln(2/3)=ln(3)=>x=(ln(3)/(ln(2/3)))~~-2.71$

$(-2.71 , 0 )$

as $x -> oo$ , $color(white)(88)-(2/3)^x+3->3$

as $x -> -oo$ , $color(white)(88)-(2/3)^x+3->-oo$

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

Graph:

enter image source here

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