We present a detailed investigation of the X-ray luminosity (Lx)-gas temperature (Tvir) relation of the complete X-ray flux-limited sample of the 64 brightest galaxy clusters in the sky (HIFLUGCS). We study the influence of two astrophysical processes, active galactic nuclei (AGN) heating and intracluster medium (ICM) cooling, on the Lx-Tvir relation, simultaneously for the first time. We determine best-fit relations for different subsamples using the cool-core strength and the presence of central radio activity as selection criteria. We find the strong cool-core clusters (SCCs) with short cooling times (< 1Gyr)to display the steepest relation (Lx ~ Tvir^{3.33}) and the non-cool-core clusters (NCCs) with long cooling times (> 7.7Gyr) to display the shallowest (Lx ~ Tvir^{2.42}). This has the simple implication that on the high-mass scale (Tvir > 2.5keV) the steepening of the Lx-Tvir relation is mainly due to the cooling of the intracluster medium gas. We propose that ICM cooling and AGN heating are both important in shaping the Lx-Tvir relation but on different length-scales. While our study indicates that ICM cooling dominates on cluster scales (Tvir > 2.5keV), we speculate that AGN heating dominates the scaling relation in poor clusters and groups (Tvir < 2.5keV). The intrinsic scatter about the Lx-Tvir relation in X-ray luminosity for the whole sample is 45.4% and varies from a minimum of 34.8% for weak cool-core clusters to a maximum of 59.4% for clusters with no central radio source. We find that after excising the cooling region, the scatter in the Lx-Tvir relation drops from 45.4% to 39.1%, implying that the cooling region contributes ~ 27% to the overall scatter. Lastly, we find the true SCC fraction to be 25% lower than the observed one and the true normalizations of the Lx-Tvir relations to be lower by 12%, 7%, and 17% for SCC, WCC, and NCC clusters, respectively. [abridged]
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