CPM Homework Help : CALC Problem 5-44

A "CHAIN" OF FUNCTIONS
Remember that composite functions are made by "chaining" simpler functions together. For example, $f(x) = 2x^2 + 3$ is made by chaining the functions $h(x) = 2x^2$ and $g(x) = x + 3$ .

It can be useful to show this idea with an arrow diagram:

Above is an arrow diagram for $h$ followed by $g$ , which would be written $f(x) = g(h(x))$ . (Notice that the "inner" function is applied first.)

Let $h(x) = \operatorname{tan }x + 5$ and $g(x) = (x + 3)^2$ . Draw an arrow diagram and determine the composite function for each $f(x)$ below. Simplifying is not required.

$f(x) = g(h(x))$
Answer (a):
$x\overset{h}{\rightarrow}\text{tan } x+5\overset{g}{\rightarrow}((\text{tan }x+5)+3)^{2}=(\text{tan }x+8)^{2}$

$f(x) = h(g(x))$
Hint (b):
Refer to example above and hint in part (a).

$f(x) = g(g(h(x)))$
Hint (c):
Refer to example above and hint in part (a).

$f(x) = h(g(g(x)))$
Hint (d):
Refer to example above and hint in part (a).