For $f\left(x\right)=\sin\left(x^2\right)$, $g ( x ) = \sqrt { x - 2 }$, and $h(x)=\frac{1}{x}$, write equations for each of the following compositions of functions and state the domains.

1. $f\left(g\left(x\right)\right)$

   Hint (a):

   Since $g$ is a square root function, the domain of $g$ is $x ≥ 2.$

   Step (a):

   $f(g(x))=\sin(\sqrt{x-2}^2)=$

2. $f\left(g\left(h\left(x\right)\right)\right)$

   Hint (b):

   $h$ is a rational function, so the denominator cannot equal $0$.

   More Help (b):

   Before writing your simplified equation, the quantity inside of the square root must be greater than or equal to $0$.

3. $h\left(f\left(g\left(x\right)\right)\right)$

   Steps (c):

   $h(\text{part (a)})=\frac{1}{\text{(part (a))}}=$

   Hint (c):

   The denominator cannot equal $0$. For what value(s) of $x$ does $\sin\left(x−2\right)=0$? Also, due to the square root in the original function, $x – 2 ≥ 0$.

   Answer (c):

   Domain: $x>2$ and $x ≠ 2+πn$, where $n$ is an integer value

4. $h\left(h\left(x\right)\right)$

   Step (d):

   $h(h(x))=\frac{1}{\frac{1}{x}}$