I have already done part a,b,c. But in d part they ask to extend the function. I can't understand how to extend it and sketch the extended function.

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Prasad
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What has this to do with ODE? A periodic continuation looks like a periodic function. – Lutz Lehmann Apr 30 '19 at 17:00
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Part b says $f(x)$ is periodic with period $2\pi$. You know what the function looks like from $-\pi$ to $\pi$. That tells you what the function looks like everywhere. – Paul Sinclair May 01 '19 at 01:34