Question: Suppose that the functions $h$ and $f$ are defined as follows.
$h(x) = x^2 - 4$
$f(x) = \frac{x}{9}$
Find the compositions $h \circ h$ and $f \circ f$.
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.)
$(h \circ h)(x) =$
$(f \circ f)(x) =$
1. **State the problem:** We are given two functions:
$$h(x) = x^2 - 4$$
$$f(x) = \frac{x}{9}$$
We need to find the compositions $(h \circ h)(x)$ and $(f \circ f)(x)$, which means:
$$(h \circ h)(x) = h(h(x))$$
$$(f \circ f)(x) = f(f(x))$$
2. **Find $(h \circ h)(x)$:**
Start by substituting $h(x)$ into $h$:
$$h(h(x)) = h\left(x^2 - 4\right)$$
Using the definition of $h$, replace $x$ by $x^2 - 4$:
$$h\left(x^2 - 4\right) = \left(x^2 - 4\right)^2 - 4$$
3. **Simplify $(x^2 - 4)^2 - 4$:**
First expand the square:
$$\left(x^2 - 4\right)^2 = (x^2)^2 - 2 \cdot x^2 \cdot 4 + 4^2 = x^4 - 8x^2 + 16$$
So:
$$h(h(x)) = x^4 - 8x^2 + 16 - 4 = x^4 - 8x^2 + 12$$
4. **Find $(f \circ f)(x)$:**
Substitute $f(x)$ into $f$:
$$f(f(x)) = f\left(\frac{x}{9}\right)$$
Using the definition of $f$, replace $x$ by $\frac{x}{9}$:
$$f\left(\frac{x}{9}\right) = \frac{\frac{x}{9}}{9} = \frac{x}{9 \cdot 9} = \frac{x}{81}$$
5. **Final answers:**
$$(h \circ h)(x) = x^4 - 8x^2 + 12$$
$$(f \circ f)(x) = \frac{x}{81}$$