Inverse-Gaussian apodized fiber Bragg gratings (IGAFBGs) are numerically studied using the transfer matrix method and fabricated by the commonly used phase-mask scanning technique in a single-step scanning process. The IGAFBG can serve as a dual-wavelength passband filter, whose wavelength spacing can be continuously tuned by introducing a tunable chirp through applying a strain gradient in principle. Also, an IGAFBG with identical dual passbands having 0.144nm wavelength spacing is experimentally achieved. We also show that an IGAFBG can act as a multipassband filter with varied free spectral ranges (FSRs), and the largest FSR variation of this IGAFBG is nearly seven times more than that in a comparable FBG pair filter. An IGAFBG with varied FSRs of approximately 16.125, approximately 12.25, approximately 8.5, and approximately 6.375GHz is fabricated. This multipassband varying-FSR IGAFBG filter can find applications in step-tunable microwave generations.