1. 1.6 Composition of Functions Thurs Sept 18
Do Now
Find the domain of

2. HW Review: p.131 #39-49 59-63 39-49) Graph in book
59) D: (-inf, inf) R: (-1, inf)
f(x) = -2 for x < 2
-5 for x = 2
4 for x > 2
61) D: (-inf, inf) R: (-inf, 1] U [2, inf)
g(x) = x for x <= -1
2 for -1 < x < 2
x for x >= 2

3. 63 63) D: [-5, 3) R: (-3, 5) h(x) = x + 8 for -5 <= x < -3

4. Composite Functions
A composite function is a function whose range is determined by the output of another function
One function is composed in another function
Ex: The DO NOW function is a square root function composed in a rational function

5. Composition of Functions
The composite function , the composition of f and g, is defined as
where x is in the domain of g, and g(x) is in the domain of f
Note: This means only the range of g(x) can be plugged into f(x)!

6. Ex
Given that f(x) = 2x – 5 and g(x) = x^2 – 3x + 8, find f(g(x)) and g(f(x))

7. Compositions and Domains
When considering the domain of a composite function, you must look at the following:
The domain of the function that is being ‘plugged in’
The domain of the final composition
Note: Just because a composition simplifies into something with no restrictions does not mean the domain has no restrictions!

8. Ex
Given that and g(x) = x – 3, find f(g(x)) and g(f(x)) and determine the domains of each

9. Ex
Given that f(x) = 1 / (x – 2) and g(x) = 5/x, find f(g(x)) and g(f(x)), and the domain of each

10. Closure
Given f(x) = 3x – 2 and g(x) = x^2 + 5, find f(g(x)) and g(f(x))
HW: p.144 #69-87 odds
Check your answers! I don’t put up the answers anymore!

11. 1.6 Decomposing Compositions Fri Sept 19
Do Now
Given and g(x) = 2 – 3x, find f(g(x)) and g(f(x)) and their domains

12. HW: p.144 #69-87 odds
Check your book

13. Decomposing a Function
In calculus we’ll need to identify which function is the ‘inside’ and the ‘outside’
This method is called decomposing

14. Ex
If h(x) = (2x – 3)^5, find f(x) and g(x) such that h(x) = f(g(x))

15. Ex
If , find f(x) and g(x) if h(x) = f(g(x))

16. Closure If , find f(x) and g(x) if h(x) = f(g(x))
HW: p.144 # odds
Quiz on Tuesday

17. Review Mon Sept 22
Do Now
For each function h(x), identify the f(x) and g(x) such that h(x) = f(g(x))

18. HW Review: p.144 #91-101

19. 1.2 1.5 1.6 Review 1.2 Domain of Functions
1.5 Finding increasing/decreasing intervals
1.5 Finding max/mins
1.5 Piecewise Functions
1.6 Function Composition
1.6 Decomposing Compositions
Full period quiz – 50 pt quiz

20. Closure What is a composition? How do you evaluate one?
quiz tomorrow!