A graph is defined in polar coordinates by $r = \cos \theta + \frac{1}{2}.$  Find the smallest $x$-coordinate of any point on this graph.
The $x$-coordinate of a point on this graph is given by
\begin{align*}
x &= r \cos \theta \\
&= \left( \cos \theta + \frac{1}{2} \right) \cos \theta \\
&= \cos^2 \theta + \frac{1}{2} \cos \theta \\
&= \left( \cos \theta + \frac{1}{4} \right)^2 - \frac{1}{16}.
\end{align*}The minimum value is then $\boxed{-\frac{1}{16}},$ which occurs when $\cos \theta = -\frac{1}{4}.$