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

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]}