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

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.

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: (-∞,∞)

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,∞)

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.

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,∞)

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

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

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.

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

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

Homework Pg. 124: 2-6 and 12-18 even - state domains when directed- and 23-27all