How do you find the inverse of #e^-x# and is it a function?

1 Answer
Mar 20, 2016

#-ln(x)#

Explanation:

We have the function

#y=e^-x#

To find its inverse, swap #y# and #x#.

#x=e^-y#

Solve for #y# by taking the natural logarithm of both sides.

#ln(x)=-y#

#y=-ln(x)#

This is a function. We knew that it would be because of the graph of #y=e^-x#:

graph{e^-x [-16.36, 34.96, -5.25, 20.4]}

There is only one #x# value for every #y# value, so its inverse will only have one #y# value for every #x# value, the definition of a function, i.e., the inverse will pass the vertical line test:

graph{-lnx [-9.77, 55.18, -7.23, 25.23]}