We explore the utility of density-functional theory (DFT) in conjunction with the transition-potential (TP) method to simulate x-ray-absorption spectra. Calculations on a set of small carbon-containing molecules and chemisorbed species show that this provides a viable option for obtaining excitation energies and oscillator strengths close to the experimental accuracy of core-valence transitions. Systematic variations in energy positions and intensities of the different spectra in the test series have been investigated, and comparison is made with respect to the static exchange-, self-consistent-field, and explicit electron-correlation methods. The choice between standard exchange-correlation functionals is shown to be of little consequence for the valence resonant, here π*, parts of the x-ray-absorption spectra, while the long-range behavior of presently available functionals is found not to be completely satisfactory for Rydberg-like transitions. Implementing a basis set augmentation technique, one finds that DFT methods still account well for most of the salient features in the near-edge x-ray-absorption spectra, save for the multielectron transitions in the near continuum, and for some loss of Rydberg structure. For clusters modeling surface adsorbates, the DFT transition potential method reproduces well the spectral compression and intensity reduction for the valence level absorption compared to the free phase, provided fairly large clusters are taken into account. While for near-edge x-ray-absorption fine-structure (NEXAFS) spectra of free molecules the DFT-TP and Hartree-Fock/static exchange methods have complementary advantages, the DFT-TP method is clearly to be preferred when using clusters to simulate NEXAFS spectra of surface adsorbates.