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) = 2x² + 3 is made by chaining the functions h(x) = 2x² 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) = 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. Homework Help ✎
1. f(x) = g(h(x))
2. f(x) = h(g(x))
3. f(x) = g(g(h(x)))
4. f(x) = h(g(g(x)))

Answer (a):

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

Hint (c):

Refer to example above and hint in part (a).

Hint (b):

Refer to example above and hint in part (a).

Hint (d):

Refer to example above and hint in part (a).