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

2 Answers
mason m
Jan 9, 2016

f^-1(x)=-lnxf−1(x)=−lnx

Explanation:

y=e^-xy=e−x

Switch the xx and yy and solve for yy.

x=e^-yx=e−y

To get the -y−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 ee.

lnx=-ylnx=−y

y=-lnxy=−lnx

In function notation:

f^-1(x)=-lnxf−1(x)=−lnx

Tony B
Jan 9, 2016

y=-ln(x)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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