1. Today in Pre-Calculus Go over homework questions Notes: Homework
Domains of combined and composition of functions
Decompositions
Homework

2. Domains of Combined Functions
Let f and g be two functions with intersecting domains. Then the algebraic combinations of f and g are defined to have a domain that consists of all the numbers that belong to both the domain of f and the domain of g.

3. Example
Let f(x) = 3x2 + 2 and g(x) = 5x – 4. Find the functions and domains for the:
(f + g)(x)
(f – g)(x)
(fg)(x)
Quotient
Domain: (-∞,∞)
Domain: (-∞,∞)
Domain: (-∞,∞)

4. Example Let and Find (fg)(x) and state its domain Domain of f(x):[0,∞)
Domain of g(x): (-∞,2)υ(2,3)υ(3,∞)
Domain of (fg)(x): [0,2)υ(2,3)υ(3,∞)

5. Domains of Composition of Functions
The domain of f◦g consists of all x-values in the domain of g that map to g(x)-values in the domain of f.
Start with domain of the inside function and include further restrictions required by the new function.

6. Example
Let f(x) = x2 – 1 and g(x) = Find f(g(x)) and state the domain.
Domain f(x) = (-∞,∞)
Domain g(x)=[0,∞)
f(g(x)) =
Domain=[0,∞)

7. Example Domain f(x)= (-∞,-1)υ(-1,∞) Domain g(x)= (-∞,1)υ(1,∞)

8. Example Domain f(x)= (-∞,-1)υ(-1,∞) Domain g(x)= (-∞,1)υ(1,∞)

9. Decomposition of Functions
Allows us to think of a complex function in terms of two or more simpler functions.
In the composition f(g(x)), view f as the outside function and g as in the inside function.

10. Example Find two function f(x) and g(x) so that h(x)=f(g(x))
The inside function is x2 + 1 and the outside function is the square root

11. Examples
Find two function f(x) and g(x) so that h(x)=f(g(x))

12. Homework
Pg. 124: 2-6 and even - state domains when directed- and 23-27all