How do you find the inverse of # f(x)=e^-x#?

2 Answers
mason m
Jan 9, 2016

#f^-1(x)=-lnx#

Explanation:

#y=e^-x#

Switch the #x# and #y# and solve for #y#.

#x=e^-y#

To get the #-y# out of the exponent, take the natural logarithm of both sides. The logarithm is the inverse function of exponentiation, so it will undo the #e#.

#lnx=-y#

#y=-lnx#

In function notation:

#f^-1(x)=-lnx#

Tony B
Jan 9, 2016

#y=-ln(x)#

#color(blue)(color(white)(....)"See explanation and graph")#

Explanation:

Let #y=e^(-x)#

Take logs both sides

#ln(y)=ln(e^(-x))#

#ln(y)=-xln(e)#

But #ln(e)=1# giving

#ln(y)=-x#

Swap the variable letters giving

#ln(x)=-y#

Multiply by (-1) giving

#y=-ln(x)#

#color(blue)("Something to think about!")#

The inverse function is a reflection of the original equation about the line # y= x#

In this case you have:
Tony B

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