Suppose that the functions $$ f $$ and $$ g $$ are defined as follows. $$ f(x)=x^{2}-9 $$ $$ g(x)=\frac{x}{2} $$ Find the compositions $$ f $$ of and $$ g \circ g $$. 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 indicate the domain.) $$ (f \circ f)(x)=\square $$

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1. To find $$ (f \circ f)(x) $$, we first apply $$ f $$ to $$ x $$ and then apply $$ f $$ to the result:
   $$
   f(x) = x^2 - 9
   $$
   Now compute:
   $$
   (f \circ f)(x) = f(f(x)) = f(x^2 - 9).
   $$

2. Substitute $$ x^2 - 9 $$ into $$ f(x) $$:
   $$
   f(x^2 - 9) = (x^2 - 9)^2 - 9.
   $$

3. Expand $$ (x^2 - 9)^2 $$:
   $$
   (x^2 - 9)^2 = x^4 - 18x^2 + 81.
   $$

4. Therefore:
   $$
   (f \circ f)(x) = x^4 - 18x^2 + 81 - 9 = x^4 - 18x^2 + 72.
   $$

5. Now, to find $$ (g \circ g)(x) $$, we apply $$ g $$ twice:
   $$
   g(x) = \frac{x}{2}
   $$
   So we have:
   $$
   (g \circ g)(x) = g\left(g(x)\right) = g\left(\frac{x}{2}\right) = \frac{\frac{x}{2}}{2} = \frac{x}{4}.
   $$

The final simplified expressions are:
$$
(f \circ f)(x)= x^4 - 18x^2 + 72 \quad \text{and} \quad (g \circ g)(x)= \frac{x}{4}.
$$
$$ (f \circ f)(x) = x^4 - 18x^2 + 72 $$ and $$ (g \circ g)(x) = \frac{x}{4} $$.

Suppose that the functions and are defined as follows.

Find the compositions of and .
Simplify your answers as much as possible.
(Assume that your expressions are defined for all in the domain of the composition. You do not have to indicate the domain.)

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Suppose that the functions and are defined as follows. Find the compositions of and . Simplify your answers as much as possible. (Assume that your expressions are defined for all in the domain of the composition. You do not have to indicate the domain.)