3.6-2 Composing, Decomposing Functions

1. 3.6-2 Composing, Decomposing Functions

2. 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

3. 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

4. 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

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

6. 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)

7. 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

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

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

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