find the exact value of the expression, if it is defined. (if an answer is undefined, enter undefined.)…

Explanation:

Step1: Recall inverse - cosine property

The function $y = \cos^{-1}(x)$ has a domain of $[- 1,1]$ and range of $[0,\pi]$. The property $\cos(\cos^{-1}(x))=x$ holds for $x\in[-1,1]$.

Step2: Check the value of $x$

Here, $x =-\frac{1}{9}$, and since $-1\leqslant-\frac{1}{9}\leqslant1$.

Step3: Apply the property

By the property $\cos(\cos^{-1}(x)) = x$ for $x\in[-1,1]$, we have $\cos(\cos^{-1}(-\frac{1}{9}))=-\frac{1}{9}$.

Answer:

$-\frac{1}{9}$