1.6 Operations on Functions and Composition of Functions

• Pg. 73 # 121 – 123, 125 – 128 Pg. 67 # 9 – 17 odd, 39 – 42 all

• #121 2L + 440 #122 l = 125, A = 125w• #123 t = 6.16 hrs #124 r = 6.91 units• #125 P(n) = 0.50n – 18.25 #126 Graph• #127 D: {0,1,2…} R: {-18.25, -17.75, -17.25,…} • #128 37 tickets• #1 (∞,∞);(∞,∞);(∞,∞);(∞,0)U(0,∞)• #3 (∞,∞);(∞,∞);(∞,∞);(∞,0)U(0,∞)• #5 (∞,∞);(∞,∞);(∞,∞);(∞, ½)U(½,∞) #7 D: (∞,4)U(4,∞) R: {-1,1}• #9 (f ◦ g)(3) = 8; (g ◦ f)(-2) = 3 #11 (f ◦ g)(3) = 9; (g ◦ f)(-2) = 66• #35 Graph #36 Graph #37 Graph #38 Graph

1.6 Operations on Functions and Composition of Functions

• The perimeter P of a rectangle is given by the equation P = 2L + 2W, where L is the length and W is the width. If the width is 200 units, then write an equation for the perimeter P as a function of the length. Find a complete graph showing how P varies with length.

1.6 Operations on Functions and Composition of Functions

Composition of Functions• Notation is given by:

• In order for a value of x to be in the domain of f◦g, two conditions must be met:– x must be in the domain of f– f(x) must be in the domain of

g

Practice• Let and

– Find and and determine their domain.

• Let and– Find and

and determine their domain.

f g x f g x

2 1f x x g x x

f g x g f x

2f x x 9g x x

f g x g f x

1.6 Operations on Functions and Composition of Functions

Composition of Functions• Notation is given by:

• In order for a value of x to be in the domain of f◦g, two conditions must be met:– x must be in the domain of f– f(x) must be in the domain of

g

Practice• Let f(x) = 2x + 1 and

g(x) = x1/2 - 2– Find and

and determine their domain.

• Let f(x) = 2x3 - 1 andg(x) = x + 5– Find and

and determine their domain.

f g x f g x f g x g f x

f g x g f x

1.6 Operations on Functions and Composition of Functions

Composition Effects on Transformations and Reflections

• Depending on what you are composing, you could just be creating a shift or reflection of a function.

• Look at what is inside thef◦g(x) to see if anything could transpire before you would consider graphing the new function.

Balloon Fun!! • A spherically shaped balloon

is being inflated so that the radius r is changing at the constant rate of 2 in./sec. Assume that r = 0 at time t = 0. Find an algebraic representation V(t) for the volume as a function of t and determine the volume of the balloon after 5 seconds.

1.6 Operations on Functions and Composition of Functions

Shadow Movement• Anita is 5 ft tall and walks at

the rate of 4 ft/sec away from a street light with it’s lamp 12 ft above ground level. Find an algebraic representation for the length of Anita’s shadow as a function of time t, and find the length of the shadow after 7 sec.

More Rectangles!!• The initial dimensions of a

rectangle are 3 by 4 cm, and the length and width of the rectangle are increasing at the rate of 1 cm/sec. How long will it take for the area to be at least 10 times its initial size?