We develop a new diagnostic method to classify galaxies into AGN hosts, star-forming galaxies, and absorption-dominated galaxies by combining the [O III]/Hbeta ratio with rest-frame U-B color. This can be used to robustly select AGNs in galaxy samples at intermediate redshifts (z<1). We compare the result of this optical AGN selection with X-ray selection using a sample of 3150 galaxies with 0.3<z<0.8 and I_AB<22, selected from the DEEP2 Galaxy Redshift Survey and the All-wavelength Extended Groth Strip International Survey (AEGIS). Among the 146 X-ray sources in this sample, 58% are classified optically as emission-line AGNs, the rest as star-forming galaxies or absorption-dominated galaxies. The latter are also known as X-ray bright, optically normal galaxies (XBONGs). Analysis of the relationship between optical emission lines and X-ray properties shows that the completeness of optical AGN selection suffers from dependence on the star formation rate and the quality of observed spectra. It also shows that XBONGs do not appear to be a physically distinct population from other X-ray detected, emission-line AGNs. On the other hand, X-ray AGN selection also has strong bias. About 2/3 of all emission-line AGNs at L_bol>10^44 erg/s in our sample are not detected in our 200 ks Chandra images, most likely due to moderate or heavy absorption by gas near the AGN. The 2--7 keV detection rate of Seyfert 2s at z~0.6 suggests that their column density distribution and Compton-thick fraction are similar to that of local Seyferts. Multiple sample selection techniques are needed to obtain as complete a sample as possible.
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