continuous signals - Understanding the frequency domain

I'm trying to understand what frequency domain is. I found general explanations on the Internet, for example:

• frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies


• Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time.


• Fourier series transform a signal from time domain to frequency domain.

But I could not find an example which shows how I can obtain the frequency domain graph using the formula of a signal.

The graphs I found are sometimes discrete like this one:

and sometimes continuous like this one:

which seems confusing.

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

It's clear how to plot its time domain graph, But how can I find its frequency domain function or graph?

Should I apply fourier transform on it?:

$$\widehat{x}(\tau)=\int_{-\infty}^\infty \cos(6\pi t)e^{-\pi t^2}e^{-2\pi t \tau i}dt$$

Does plotting this new function gives the frequency domain graph? (but it is complex valued, how can it be plotted?)

Or should I find the Fourier series of $x(t)$ and plot the series coefficients discretely?

So my only question is: How can I mathematically obtain the the frequency domain function (and then plot it to get the frequency domain graph) using the formula of $x(t)$?