Inverse of binary funcation

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Is it possible to get the inverse function $f^{-1}(y)$ from the one below? Don't really know how to handle $\mathbf{1}_{\{ p(k)\leq x\}}$. $$ f(x)=\sum_{k=0}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\mathbf{1}_{\{ p(k)\leq x\}} $$ Thank you.