Solve equation when function is min

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I have the following function $$ Q=F(L,K) = min (L,K) , L,K>0 $$

And I would like to solve the equation for K. So as to make the graph L,K

Could you help please ?

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\begin{equation} K= \begin{cases} Q: L>Q \\ L+x:x>0,L=Q \\ \end{cases} \end{equation}

If $L<Q$, the equation has no solution for $K$.