3.6-2 Composing, Decomposing Functions

Recall from yesterday, we can make a combination of functions through using the basic operations +, -, /, x Now, we can compose a new function by essentially “embedding” one function into another

Composition Composition = the composition of a function f and g, denoted as , is defined as f(g(x)) Read as “f of g,” or “fog” Not the same as g(f(x)) Just like yesterday, we can evaluate as a point, OR as a new function with a new defined domain

After we compose a function, we can then use it evaluate new function values (f ○ g)(x) is the new function (f ○ g)(5) means to replace every x, in the composed function, with a 5

Example. If f(x) = 2x – 3, and g(x) = x + 5, find: A) (f ○ g)(6) B) (f ○ g)(x)

Example. If f(x) = x2 + 2, and g(x) = x, find the formulas and state the domains for: A) (f ○ g)(x) B) (g ○ f)(x)

Decomposition On the other hand, we can decompose = break down a function into simpler/separate functions Best to work “inside-out” Look at the argument, and determine what the function is doing to the argument Portion inside the main function

Example. Decompose the function f(x) = (x4 + 1)3 into two functions. What is inside the argument? What is on the outside?

Example. Decompose the function f(x) =. into two functions Example. Decompose the function f(x) = into two functions. And, three functions.

Assignment Pg. 270 #31-65 odd Just find the formulas for fog on 41-53.