# inverse functions

• September 17th 2008, 10:35 PM
ericchung912
inverse functions
how do you find the inverse of f(X)=2^ -x ?

• September 17th 2008, 10:59 PM
Soroban
Hello, ericchung912!

Quote:

How do you find the inverse of: . $f(x)\:=\:2^{-x}$ ?

We have: . $f(x) \;=\;2^{-x}$

Replace $y$ with $f(x)\!:\;\;y \;=\;2^{-x}$

Switch x's and y's: . $x \;=\;2^{-y}$

Solve for y: . $2^{-y} \;=\;x$

Take logs: . $\ln\left(2^{-y}\right) \;=\;\ln(x) \quad\Rightarrow\quad -y\!\cdot\!\ln(2) \;=\;\ln(x) \quad\Rightarrow\quad y \;=\;-\frac{\ln(x)}{\ln(2)}$

Therefore: . $f^{-1}(x) \;=\;-\frac{\ln(x)}{\ln(2)}$