jmc

To find $f^{-1}(7),$ we try solving $f(x)=7$ on each piece.

If $x+3=7$, then $x=4,$ which satisfies $x<20.$ If $2x-2=7,$ then $x=\frac{9}{2},$ which does not satisfy $x\ge 20,$ so $f^{-1}(7)=4.$

Similarly, $x+3=46,$ then $x=43,$ which does not satisfy $x<20.$ If $2x-2=46,$ then $x=24,$ which satisfies $x\ge 20,$ so $f^{-1}(46)=24.$

Hence, $f^{-1}(7)+f^{-1}(46)=4+24=\boxed{28}.$