Question

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

Answer 1

The domain is `x in RR`. The range is `y in (0, +oo)`

Explanation:

The function is

`y=2^(-x)=1/2^x`

`lim_(x->-oo)y=lim_(x->-oo)1/2^x=oo`

`lim_(x->+oo)y=lim_(x->+oo)1/2^x=0`

`AA x in RR, 1/2^x>0`

The domain of `y` is `x in RR`

`2^x=1/y`

`ln(2^x)=ln(1/y)`

`xln2=ln(1/y)=ln1-lny=-lny`

`x=-1/ln2lny`

Therefore,

`y in (0, +oo)`

The range is `y in (0, +oo)`

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