Question

How do you find the domain and range of `e^(-4t)`?

Answer 1

Domain: All Reals
Range: Y>0

Explanation:

Start by finding the Domain (t values) and then use that information to solve for the Range (y values)

for `e^(-4t)` look for the number t cannot be.

Look for things such as:
-Dividing by 0
-Taking a root of a negative number
-Taking the `ln` of a negative number

`e^(-4t) = 1/(e^(4t))`

Now the only issue is that the denominator cannot equal 0.
But `e` (being a positive number) to any power will be a number greater than 0 and so there are no exceptions for `t`.

Therefore the Domain is all real numbers.

The Range would then be the results of all the values of `t`.

When t is really really big the equation gets closer and closer to 0.
`(1/10000000 = 0.0000001)` As `t` gets bigger the the result gets closer and closer still to zero, but never equals zero. So its always positive as `t` gets larger.

When `t` is small (less than 0) you get `e^-(4(-t))` or simply `e^(4t)`
So as `t` increases the result goes to infinity. So again we get all positive numbers.

Therefore the range is all values greater than 0.

You can also see this graphically.

graph{e^(-4x) [-10, 10, -10,10]}