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

Question

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

Anna Allen · Accepted Answer

#-ln(x)#

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]}

Chloe Alexander · Answer

undefined To find the inverse of $e^{-x}$, first set $y = e^{-x}$. Then, switch the roles of $x$ and $y$ and solve for $y$:

$$x = e^{-y}$$

Take the natural logarithm of both sides:

$$ \ln(x) = \ln(e^{-y}) = -y $$

Multiply both sides by -1:

$$ -\ln(x) = y $$

Therefore, the inverse function of $e^{-x}$ is $y = -\ln(x)$, which is commonly denoted as $f^{-1}(x) = -\ln(x)$.

Yes, $e^{-x}$ is a function, and its inverse $f^{-1}(x) = -\ln(x)$ is also a function. Both are continuous functions defined for all real numbers.