= Solution
For a real nonzero $b$, put $s=bt$ in the <Fourier transform>. Reversing the integration limits when $b<0$ produces the absolute value:
$$
\boxed{\mathcal F[f(b\,\cdot)](\omega)=\frac1{|b|}\widetilde f\left(\frac\omega b\right)}.
$$
This is the <scaling property of the Fourier transform>. If $b=0$ and $f(0)$ exists, the transformed function is constant in time; its transform is $2\pi f(0)\delta(\omega)$ in the sense of <tempered distributions>, rather than an ordinary convergent Fourier integral.
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