Let $f(t)$ be continuous time signal and let $F(w)$be its Fourier Transform defined by

$F(\omega )=\displaystyle{}\int_{-\infty }^{\infty }f(t)e^{-j\omega t} dt$

define $g(t)$ by

$g(t)=\displaystyle{}\int_{-\infty }^{\infty }F(u)e^{-jut} du$

What is the relationship between $f(t)$ and $g(t)?$

1. $g(t)$ would always be proportional to $f(t)$.
2. $g(t)$ would be proportional to $f(t)$ if $f(t)$ is an even function.
3. $g(t)$ would be proportional to $f(t)$ only if $f(t)$ is a sinusoidal function.
4. $g(t)$ would never be proportional to $f(t)$.

edited