As the title suggests, I would like to show that for $\phi \in C^{0}(\mathbb{R}^{n})$ with $(1+|x|)^{N} \phi(x)$ bounded for some $N>n$, $$\overline{\mathscr{F}\phi(\xi)}= (2\pi)^{n}(\mathscr{F^{-1}\overline{\phi}})(\xi),$$ where the bar denotes complex conjugation.
I have recently learnt the basics of Fourier transform and as such I am still not very familiar with the proving type of questions. This is an optional question in my tutorial and I would be extremely grateful if someone could give me a hint or show me how it could be done.
The condition implies that $\phi\in L^{1}$ and hence the Fourier transform of $\phi$ makes sense, for the formula, one has \begin{align*} \overline{\mathcal{F}\phi(\xi)}&=\overline{\int\phi(x)e^{-2\pi ix\cdot\xi}dx}\\ &=\int\overline{\phi(x)}e^{2\pi ix\cdot\xi}dx\\ &=\int\overline{\phi}(x)e^{2\pi ix\cdot\xi}dx\\ &=\mathcal{F}^{-1}\overline{\phi}(\xi), \end{align*} the additional constant depends on the context.