a question about Fourier transforms

Question

I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ?

$\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$

$if -t=x\to -dt=dx$

$\int^\infty_{-\infty}f(x).e^{iwx}(-dx)$

Answer 1

$\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$

$\text{ if } -t=x\to -dt=dx$

when $t=-\infty \to x= + \infty$

when $t=+\infty \to x= - \infty$

$\int^{-\infty}_{+\infty}f(x)e^{iwx}(-dx)=\int^\infty_{-\infty}f(x)e^{-i(-w)x}dx=\int^\infty_{-\infty}f(t)e^{-i(-w)t}dt=\mathcal F(f(t))(-w)$