Apply Fourier Transform of a function to find Fourier Transform of another function

Question

My problem is: Finding the Fourier Transform of the function below:

enter image description here and use it to find the Fourier Transform of the function that have the graph as below:

enter image description here

I know that to find that, I have to separate the function into 3 parts (from -6 to -2, from -2 to 2, from 2 to 6), but can I calculate all of them in the interval (-2,2) (For example, find the Fourier Transform of a function that has the same shape of the part in (-6,-2) in (-2,2)). Is it okay?

Answer 1

Hint:

If you can find the transform of $f(x)$ what can you say about $f(x+b)$? Also, Fourier transform is linear, so what about the transform of $f(x)+af(x+b)+cf(x-d)$?(a,b,c,d are constants)

Answer 2

It depends on your definition of your Fourier transform. If your Fourier transform scales by $\pi$ or $2\pi$ in the exponential function, then the periodicity of the complex exponential will allow you to easily translate between the intervals given.