Clear[e, tT, t, omega]

e[t_] := E^(-t^2/tT^2) Cos[ omega t ]

Plot[ Block[{tT=1,omega=10},e[t]],{t,-2.5,2.5}]

ft[ω_] = Distribute[FullSimplify[FourierTransform[ e[t], t, ω]]]

Plot[ Block[{tT=1,omega=10},ft[ω]],{ω,-20,20}]

Plot[1.1Tanh[x] + 1.3, {x, -10, 10}]

Plot[{ x^2, 2 x^2, 30 E^{-x^2/2}}, {x, -5, 5}]