Answer
$f^{-1}(x)=\displaystyle \frac{\log_{2}x}{2}$
Work Step by Step
To find the inverse of y=f(x):
1. swap x and y in the expression of the function
2. solve for y
3. replace $y $ with $f^{-1}(x)$
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$y=2^{2x}$
$ x=2^{2y}\qquad$... apply $\log_{2}$ to both sides
$\log_{2}x=2y\qquad/\div 2$
$y=\displaystyle \frac{\log_{2}x}{2}$
$f^{-1}(x)=\displaystyle \frac{\log_{2}x}{2}$
