1.6 Composition of FunctionsThurs Sept 18

Do NowFind 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 = 24 for x > 2

• 61) D: (-inf, inf) R: (-inf, 1] U [2, inf)g(x) = x for x <= -1

2 for -1 < x < 2x for x >= 2

63

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

3 for -3 <= x <= 1

3x – 6 for 1 < 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 thatand 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 CompositionsFri 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 odds1.2 1.5 1.6 Quiz on Tuesday

1.2 1.5 1.6 ReviewMon 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!