Question

What is the domain and range of `y=2e^(-x)`?

Answer 1

Domain: `RR`
Range: `RR^+`

Explanation:

My advice is to always graph a function so you can see what it looks like:

graph{2e^-x [-10, 10, -5, 5]}

Just from looking:

Domain: `RR`
Range: `RR^+`

Does the graph touch zero? How high does it go? Let's be a bit more rigorous and find out by looking at the behaviour as `x` tends to `+-oo`:

`lim_(x->oo)(2e^-x)=2e^(-oo)=2/e^oo=2/oo=0`

`lim_(x->-oo)(2e^-x)=2e^(--oo)=2e^oo=oo`

This tells us that the graph asymptotes towards `y=0` as `x` goes to positive infinite, which means that it approaches but never touches zero; and it tends to `y=oo` as `x` goes towards negative infinite. Therefore, our range is positive real numbers, just as we thought.

There are no restrictions on the values `x` can take, so the domain is all real numbers.