Both examples are $L^1_{loc}$ function which are bounded at infinity, thus are tempered distributions, that is continuous linear forms on the Schwartz space $\mathscr S$ of rapidly decreasing function. The definition of the Fourier transform on $\mathscr S'$ is $$ \langle \hat u,\phi\rangle_{\mathscr S',\mathscr S}= \langle u,\hat \phi\rangle_{\mathscr S',\mathscr S}. $$ From this definition, you get for instance that the Fourier transform of $x^{-1/2}1_{\mathbb R_+}(x)$ is an homogeneous distribution of degree $-1/2$.
Bazin.