Assume that we previously calculated the Fourier series for $f(x) = \dfrac{-\pi}{2}, -\pi \le x < 0$ and $f(x) = \dfrac{\pi}{2}, 0 \le x \le \pi$, and $f(x) = \dfrac{-x}{2}, -\pi \le x < 0$ and $f(x) = -\dfrac{x}{2}, 0 \le x \le \pi$.
If we were asked to then find the Fourier series for the function defined by $f(x) = \dfrac{-\pi}{2} - \dfrac{x}{2}, -\pi \le x < 0$ and $f(x) = \dfrac{\pi}{2} - \dfrac{x}{2}, 0 \le x \le \pi$, could we simply add the results of the two previously calculated Fourier series?
And would I be correct in thinking that this is principle is the "linearity" of Fourier series?
I would greatly appreciate it if people could please take the time to clarify this.
Yes.
If you write down the integrals for the fourier coefficients of the new f(x), you can use linearity of the integral to get it in terms of the coefficients the two previous functions you found the fourier series of.