State the domain of each of the following functions. Note: The functions mentioned in parts (c) and (d) refer to those in parts (a) and (b).

1. $f(x)=\frac{1}{x+2}$

   Hint (a):

   $\text{Visualize }f(x).\text{ It is a horizontal shift of }y=\frac{1}{x},$

   which has a vertical asymptote at $x = 0$. $f(x)$ also has a vertical asymptote. Where is it, and how will the vertical asymptote restrict the domain?

2. $g(x)=\sqrt{x-4}$

   Hint (b):

   $\text{Visualize }g(x).\text{ It is a horizontal shift of }y=\sqrt{x},$

   which does not exist for negative values of $x$. What can you conclude about $g(x)$.

3. $h(x) = f(g(x))$

   Hint (c):

   $\text{Find the domain of }\left ( \frac{1}{\sqrt{x-4}+2} \right )$

   More Help (c):

   Both the square root and the denominator have restricted domains. Combine.

4. $k(x) = g(f(x))$

   Hint (d):

   Refer to hints in part (c).