Answer
(a) $g(2,-1)=1$
(b) Domain: $\mathbb{R^2}.$
(c) Range: $[-1,1]$.
Work Step by Step
(a) By direct calculation
$$g(2,-1)=\cos(2+2\cdot(-1))=\cos0=1.$$
(b) The argument of cosine can be any real number so $x+2y$ can take any real value. Also, any real $x$ and $y$ can be taken into consideration because using the formula $x+2y$ will just produce another real number which is a 'legal' argument for cosine. So the domain is $$\mathcal{D}=\mathbb{R}×\mathbb{R}=\mathbb{R}^2.$$
(c) Cosine can only take values between $-1$ and $1$. Since the argument of the domain will cover all real numbers, the whole range of the cosine will be covered as well so we get for the range
$$\mathcal{R}=[-1,1].$$
