To evaluate the function \( f(x) = \cos(-x) \) at \( x = -\frac{3\pi}{4} \), we first substitute \( x \) with \(-\frac{3\pi}{4}\):
\[ f\left(-\frac{3\pi}{4}\right) = \cos\left(-\left(-\frac{3\pi}{4}\right)\right) = \cos\left(\frac{3\pi}{4}\right) \]
Next, we calculate \( \cos\left(\frac{3\pi}{4}\right) \). The angle \( \frac{3\pi}{4} \) is in the second quadrant, where the cosine is negative. Specifically:
\[ \frac{3\pi}{4} = \pi - \frac{\pi}{4} \]
We know that:
\[ \cos\left(\frac{3\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) \]
Since \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), we find:
\[ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]
Thus,
\[ f\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]
Therefore, the final answer is:
\[ \boxed{-\frac{\sqrt{2}}{2}} \]