The impact of feedback from active galactic nuclei (AGNs) on the cosmological evolution of the large-scale structure is a long-studied problem. However, it is still not well understood how the feedback energy couples to the ambient medium to influence the properties of AGN host galaxies and dark matter halos. In this work we investigate different modes of AGN feedback and their effect on the surrounding medium by probing the diffuse X-ray emission from the hot gas inside galaxy groups and clusters. For this purpose, we use the cosmological hydrodynamic simulation SIMBA to theoretically calculate the X-ray emission from simulated galaxy clusters/groups with the help of the Astrophysical Plasma Emission Code. We also perform synthetic observations of these systems with the Chandra X-ray telescope using the ray-tracing simulator Model of AXAF Response to X-rays. Our results show that in addition to the radiative wind mode of feedback from the AGNs, jet and X-ray modes of feedback play significant roles in suppressing the X-ray emission from the diffuse gas in the vicinity of the black hole. Our mock observational maps suggest that the signatures of AGN feedback from high-redshift objects may not be detected with the instrumental resolution of current X-ray telescopes like Chandra, but provide promising prospects for detection of these features with potential X-ray missions such as Lynx.