Question

How do you find the domain and range of `f(x) = 3(1/2)^x`?

Answer 1

Domain: `x in RR`
Range: `f(x) > 0`

Explanation:

We can first sketch the function to enable us to determine the domain and range:

graph{3*(1/2)^x [-9.16, 10.84, -2.12, 7.88]}

I am assuming you are aware of how to sketch this function

Now we can consider the domain:

The fucntion is defined for all values of `x` ie, it will always output `f(x)` where `f(x)` is real, or we can write as `AA x, f(x) in RR`

`=> x in RR`

Now we can find the range:

We see that there is a asymptote at `y = 0` we the function `f(x)` gets closer to `0` as `x-> oo` but will never touch `y=0`

But then `f(x)` can take on all the other positive real numbers:

`=> f(x) > 0`