We solve the one-dimensional nonlinear Schrödinger equation for an attractive delta-function potential at the origin, and obtain the bound state in closed form as a function of the nonlinear exponent α. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with α, becoming an almost completely extended state when At α=2, the bound state suffers a discontinuous change to a delta function-like profile. A further increase of α increases the width of the probability profile, although the bound state is stable only for α<2. The transmission of plane waves across the potential increases monotonically with α and is insensitive to the sign of the opacity Ω.
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