Answer
$f(x)$ has vertical asymptotes at every value of x such that $x=k + \dfrac{1}{2}$ where $k$ is an integer.
Work Step by Step
$f(x)=\tan{\pi x}=\dfrac{\sin{\pi x}}{\cos{\pi x}}.$
Vertical asymptotes occur when the denominator alone is $0\to$
$\cos{\pi x}=0\to \pi x=\dfrac{\pi}{2}+\pi k\to x=k+\dfrac{1}{2}$ where k is an integer.
