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We study the numerical solutions to semi-infinite-domain two-point boundary value problems and initial value problems. A smooth, strictly monotonic transformation is used to map the semi-infinite domain 𝑥∈[0,∞) onto a half-open interval 𝑡∈[−1,1). The resulting finite-domain two-point boundary value problem is transcribed to a system of algebraic equations using Chebyshev-Gauss (CG) collocation, while the resulting initial value problem over a finite domain is transcribed to a system of algebraic equations using Chebyshev-Gauss-Radau (CGR) collocation. In numerical experiments, the tuning of the map 𝜙∶[−1,+1)→[0,+∞) and its effects on the quality of the discrete approximation are analyzed.