This is a recommended problem for an engineering analysis course. I believe I've forgotten something required to complete it.
$Given\:F(\omega)=\mathscr{F}\{f(x)\}=\int_{-inf}^{inf}f(x)e^{-j\omega x}dx$
$Prove\:that\:\mathscr{F}\{f(x-x_{o})\}=F(\omega)e^{-j\omega x_{0}}$
I was instructed to use what is below to help solve the problem, but wasn't sure how it proved helpful.
$x^{'}=x-x_{0}\Longrightarrow x=x^{'}+x_{0}$
$\mathscr{F}[x-x_{0}]=\int_{-inf}^{inf}f(x-x_{0})e^{-j\omega x}dx$
$$ \begin{aligned} \mathscr{F}\left[f(x-x_0)\right] &= \int_{-\infty}^{+\infty}f(x-x_0)e^{-j\omega x}dx = \left[ \begin{array}{c} x' = x-x_0 \Leftrightarrow x = x'+x_0 \\ dx = dx' \\ x \rightarrow - \infty \Leftrightarrow x' \rightarrow - \infty \\ x \rightarrow + \infty \Leftrightarrow x' \rightarrow + \infty \end{array} \right] = \\ &= \int_{-\infty}^{+\infty}f(x')e^{-j\omega(x'+x_0)}dx' = \\ &= e^{-j\omega x_0}\int_{-\infty}^{+\infty}f(x')e^{-j\omega x'}dx'= \left[x:= x'\right] = \\ &= e^{-j\omega x_0}\underbrace{\int_{-\infty}^{+\infty}f(x)e^{-j\omega x}dx}_{\mathscr{F}\left[f(x)\right]} = \\ &= e^{-j\omega x_0}F(\omega) \end{aligned} $$