If $f(x) = \cos^{-1}(x)$, this represents the inverse cosine function, also written as $f(x) = \arccos(x)$.

To find $f\left(\frac{1}{2}\right)$, we evaluate:

$f\left(\frac{1}{2}\right) = \arccos\left(\frac{1}{2}\right)$

The value of $\arccos\left(\frac{1}{2}\right)$ is the angle whose cosine is $\frac{1}{2}$, which is:

$f\left(\frac{1}{2}\right) = \frac{\pi}{3}$

So, the answer is: $f\left(\frac{1}{2}\right) = \frac{\pi}{3}$

Would you like more details on inverse trigonometric functions, or have any questions?

1. How do we evaluate other values for $f(x) = \arccos(x)$?
2. What are the properties of the inverse cosine function?
3. How is the range of $\arccos(x)$ determined?
4. What is the relationship between $\arccos(x)$ and $\arcsin(x)$?
5. How do we find the derivatives of inverse trigonometric functions?

Tip: Remember that $\arccos(x)$ is defined only for values of $x$ in the range $[-1, 1]$.