Answer
$\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}-1\leq y\leq x^{2}+1$ $\}$
Work Step by Step
The domain of f depends on the domain of $\cos^{-1}t$,
which is the range of $\cos t,$ which is $[-1,1].$
So f will be defined only for those $(x,y)$ for which
$1\leq y-x^{2}\leq 1,$
$x^{2}-1\leq y\leq x^{2}+1$
This is the region between the curves $\left\{\begin{array}{l}
y=x^{2}-1\\
y=x^{2}+1
\end{array}\right..$
The curves themselves are included, so they are graphed with a solid line.
Domain: $\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}-1\leq y\leq x^{2}+1$ $\}$
