To evaluate the function \( f(x) = \cos(-x) \) at \( x = \frac{4\pi}{3} \), we proceed as follows:
First, substitute \( x \) into the function:
\[ f\left(\frac{4\pi}{3}\right) = \cos\left(-\frac{4\pi}{3}\right) \]
Using the property of cosine that \( \cos(-\theta) = \cos(\theta) \), we get:
\[ \cos\left(-\frac{4\pi}{3}\right) = \cos\left(\frac{4\pi}{3}\right) \]
Next, we need to find \( \cos\left(\frac{4\pi}{3}\right) \). The angle \( \frac{4\pi}{3} \) is located in the third quadrant, where the cosine function is negative.
To find the reference angle, we can subtract \( \pi \) (which is \( \frac{3\pi}{3} \)) from \( \frac{4\pi}{3} \):
\[ \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3} \]
Now we can evaluate \( \cos\left(\frac{4\pi}{3}\right) \):
\[ \cos\left(\frac{4\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) \]
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \), we have:
\[ \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \]
Therefore, we conclude:
\[ f\left(\frac{4\pi}{3}\right) = \cos\left(-\frac{4\pi}{3}\right) = -\frac{1}{2} \]
Thus, the final result is:
\[ \boxed{-\frac{1}{2}} \]