Explore: Decomposition of Functions Solutions

Decomposition of Functions Solutions

To mathematically decompose a function, work backward to pull apart a composition to create two unique functions . 
It is possible for there to be more than one way to decompose functions.
 If you are unsure that your functions are correct, put them back together to see if the result is the given function.

Try to think simply. While a composition can be complicated, the goal of decomposing functions is to think backward, asking yourself, “How did this result come about?”

Example 7

Determine two possible pairs of functions, p(x) and m(x), that form the given composition.

$[p \circ m] (x) = w (x)$ , when $w (x) = 9 \sqrt[3]{x - 2} + 6$

First Pair

Sample:
$p (x) = 9 x + 6 m (x) = \sqrt[3]{x - 2}$

Second Pair

Sample:
$p (x) = 9 \sqrt[3]{x} + 6 m (x) = x - 2$

Note

Your work may result in functions that are not shown here. To check your work, determine if the combination of your functions results in the given composition.