Today in Precalculus

1. Today in Precalculus • Go over homework • Notes: • 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) 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 domainof the inside functionand 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,∞) Domain: (-∞,-1)υ(-1,1) υ(1,∞)

8. Example Domain f(x)= (-∞,-1)υ(-1,∞) Domain g(x)= (-∞,1)υ(1,∞) Domain: (-∞,-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 12-18 even - state domains when directed- and 23-27all