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

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

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

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

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)!

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

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!

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

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

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!

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

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

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

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

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

Closure If , find f(x) and g(x) if h(x) = f(g(x)) HW: p.144 #91-101 odds 1.2 1.5 1.6 Quiz on Tuesday

1.2 1.5 1.6 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))

HW Review: p.144 #91-101

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

Closure What is a composition? How do you evaluate one? 1.2 1.5 1.6 quiz tomorrow!