Suppose that the functions $g$ and $h$ are defined as follows. \[ \begin{array}{l} g(x)=\frac{2}{x}, x \neq 0 \\ h(x)=x^{2}-4 \end{array} \] Find the compositions $g \circ g$ and $h \circ h$. Simplify your answers as much as possible. (Assume that your expressions are defined for all $x$ in the domain of the composition. You do not have to \[ \begin{array}{l} (g \circ g)(x)= \\ (h \circ h)(x)= \end{array} \]

Step 1 ：We are given two functions, \(g(x) = \frac{2}{x}\) and \(h(x) = x^2 - 4\). We are asked to find the compositions \(g \circ g\) and \(h \circ h\).

Step 2 ：The composition of a function is the process of applying one function to the results of another. So, for \(g \circ g\), we substitute \(g(x)\) into itself, giving us \(g(g(x))\). Similarly, for \(h \circ h\), we substitute \(h(x)\) into itself, giving us \(h(h(x))\).

Step 3 ：Substituting \(g(x)\) into itself, we get \(g(g(x)) = g\left(\frac{2}{x}\right) = \frac{2}{\frac{2}{x}} = x\).

Step 4 ：Substituting \(h(x)\) into itself, we get \(h(h(x)) = h\left(x^2 - 4\right) = \left(x^2 - 4\right)^2 - 4\).

Step 5 ：So, the composition \(g \circ g\) is \(x\) and the composition \(h \circ h\) is \((x^2 - 4)^2 - 4\).

Step 6 ：Final Answer: \( (g \circ g)(x) = \boxed{x} \) and \( (h \circ h)(x) = \boxed{(x^2 - 4)^2 - 4} \)