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Given $f(x)=(\frac{1}{2})^x$, determine $f^{-1}$ in the form $y=\dots$
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The inverse of a function can be thought of as its reverse. Instead of putting $x$ values in to obtain $y$ value outputs, we reverse the machine and put $y$ values in to get $x$ outputs. This means that we have to replace all the $x$'s with $y$'s and vica versa.

So if $f(x)=\left( \frac{1}{2} \right)^x$ we can rewrite this as $y=\left( \frac{1}{2} \right)^x$.

Therefore $f^{-1}$ will be $x=\left( \frac{1}{2} \right)^y$.

Now we need to get the equation into $y=$ form. For this we need to use logs. We know that if $a=B^c$ then $c=log_{a}B$.

Therefore $y=log_{\frac{1}{2}}x$.

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