Summary: A lattice point \((r,s)\in\mathbb{N}^2\) is said to be visible from the origin if no other integer lattice point lies on the line segment joining the origin and \((r,s)\). It is a well-known result that the proportion of lattice points visible from the origin is given by \(1/\zeta(2)\), where \(\zeta(s)=\smash{\sum_{n=1}^\infty}1/n^s\) denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika generalized the notion of lattice-point visibility by saying that for a fixed \(b\in\mathbb{N}\) a lattice point \((r,s)\in\mathbb{N}^2\) is \(b\)-visible from the origin if no other lattice point lies on the graph of a function \(f(x)=mx^b\), for some \(m\in\mathbb{Q}\), between the origin and \((r,s)\). In their analysis they establish that for a fixed \(b\in\mathbb{N}\) the proportion of \(b\)-visible lattice points is \(1/\zeta(b+1)\), which generalizes the result in the classical lattice-point visibility setting. In this paper we give an \(n\)-dimensional notion of \(\mathbf{b}\)-visibility that recovers the one presented by Goins et. al. in two dimensions, and the classical notion in \(n\) dimensions. We prove that for a fixed \(\mathbf{b}=(b_1,b_2,\ldots,b_n)\in\mathbb{N}^n\) the proportion of \(\mathbf{b} \)-visible lattice points is given by \(1/\zeta \left(\sum_{i=1}^n b_i \right)\). Moreover, we give a new notion of \(\mathbf{b}\)-visibility for vectors \[ \mathbf{b}=(b_1/a_1,b_2/a_2,\ldots,b_n/a_n)\in(\mathbb{Q}\backslash\{0\})^n, \] with nonzero rational entries. In this case, our main result establishes that the proportion of \(\mathbf{b}\)-visible points is \(1/\zeta\left(\sum_{i\in J}|b_i|\right)\), where \(J\) is the set of the indices \(1\leq i\leq n\) for which \(b_i/a_i<0\). This result recovers a main theorem of Harris and Omar for \(b\in \mathbb{Q}\backslash\{0\}\) in two dimensions, while showing that the proportion of \(\mathbf{b}\)-visible points (in such cases) only depends on the negative entries of \(\mathbf{b}\).