Is the Fourier transform of $f(x)$ the exact same as the Fourier transform of $f(t)$ when $t$ is related to $x$? For example if $t = x + 5$, would their Fourier transforms be the exact same?
Isn't this expression always true:
$$\int\limits_{-\infty}^{\infty}f(x)e^{-i \omega x}dx = \int\limits_{-\infty}^{\infty}f(t)e^{-i \omega t}dt$$