Understanding the frequency domain
In simple terms, Image shown here speaks for itself.

Before speaking about Fourier Transform black magic lets understand idea behind it. Work of the Mathematician Joseph Fourier demonstrated any arbitrary periodic (this is important) signal can be decomposed into bunch of sine-waves at with their corresponding amplitude and relative phases. 

Essentially, he meant basic component to build any signal is a sine-wave. In other words you can build any arbitrary signal exactly if you have information about how many different frequencies are present, amplitude of the each sine wave, and there relative phases. This job can be done by inverse Fourier transform. And as should be clear by now, Fourier Transform does the reverse give it composed time domain signal and it will tell you sine-waves amplitude for corresponding frequencies and there relative phases. Hence, the output is a complex. Which makes sense in polar co-ordinates, which gives you amplitude and phase. 

E.G. When you strum a guitar strings, it vibrates at multiple frequencies with different amplitude which makes up a beautiful sound, otherwise you know how a single frequency sounds(watch this :https://www.youtube.com/watch?v=qNf9nzvnd1k). one with the highest amplitude is known as tone.

So, as you can see in the figure attached here, Fourier transform tells you exactly the same information.


  Suppose I have a signal $x:\Bbb R\to \Bbb R$ defined by the formula
  $x(t) = \cos(6\pi x)e^{-\pi x^2}$.


For to obtain Fourier transform of the function you specified it should be function of time (t). So, I will just re-write your function above slightly differently replace x with t, since x(t).

$x(t) = \cos(6\pi t)e^{-\pi t^2}$.

Also, Take a look at this,
http://www.thefouriertransform.com/pairs/rightSidedSinusoids.php

Hope this helps.