Problem identifying the analytic expression of such determined signal
The simple answer is “Linearity”

The Fourier Transform is linear.  $g(t)$ can be expressed as a sum of signals that have known, as in there are tables of transform pairs, in most books, probably also in your course notes, along with the shift and scale properties. Look at the tables you have and find entries that you can add to get $g(t))$.  You already found rect().  Look for a triangle piece.  

A full triangle is the convolution of 2 rects.  A half triangle is a full triangle multiplied by a step.