How to prove $\mathrm{supp} ~ R(t)\in \{|x|<t\}$

Question

Let $R(t)$ define as $$R(t):=F^{-1} \left(\frac{\sin |\xi|t}{|\xi|} \right),$$ how to show that $$\mathrm{supp} ~ R(t)\in \{|x|<t\},$$ where $F$ is the Fourier transform.

Answer 1

Hint

Let $f(x)=0$ when $x\notin(0,1)$ and $f(x)=1$ for $x\in(0,1)$.

What is the Fourier transform of $f$?

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Adding some details. $$\hat{f}(t)=\int_{-\infty}^\infty f(x)e^{-itx}dx=\int_{0}^1 e^{-itx}dx=\frac{e^{-itx}-1}{-it}$$ Do you see how to continue?