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Given $f(x)=(\frac{1}{2})^x$, determine $f^{-1}$ in the form $y=\dots$
in Mathematics by Wooden (162 points) | 20 views

1 Answer

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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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