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

1 Answer
Jul 1, 2017

Answer:

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.