Calculus Unit 1, Lesson 2: Composite Functions Objectives The students will be able to: - Evaluate composite functions using all representations Simplify composite functions

Materials and Handouts Homework - Warm-Up - Answers to homework #1 - Keynote and notes template - Tic Tac Toe grids - Homework #2

#1-2 1) Composite Functions 2) Practice Skills Test (#1-1)

Time Activity 20 min Homework Check / Warm-up

- Warm up - Students check their answers to the homework and correct possible mistakes - Students are working on the warm-up:

o Find domain and range of a graph o Evaluate various inputs for the graph, and an equation

25 min Lecture / Activity - Warm-up - Main concept: Composite Functions: f(g(x)) - Arrow maps of f and g. Evaluate various inputs into f that require operations to be

performed first.. - Composition problem. Give groups time to see if they can figure out how to evaluate

the problem before explaining. Repeat with other examples, including one that can’t be determined, and one where the outside function is g.

- Repeat the process, using a graph. - Repeat the process, using equations. - Independent Practice: Give several independent practice problems in equation form. - Show the functions f(x) = -3x + 4 and g(x) = 1 – 5x. Ask students to evaluate: f(-5);

f(g(3)); f(Δ); f(x + 2); f(g(x)). Each time, show how the input is substituted in for the x-value in f(x). In the last example, substitute in g(x) to show that f(g(x)) = -3(g(x)) + 4 before simplifying.

- Show the functions f(x) = x2 – 2x + 3 and g(x) = 4x – 2. Simplify both f(g(x)) and g(f(x)). Point out that you get a different result depending on the order; composition is not commutative.

- Show what happens when one of the functions is a constant. 20 min Classwork

Give each group a Relay Race checker. For each round, when all the members of the group have their sheets completed and correct, the group earns a stamp and moves on to the next round. At the end of the lesson, each member of the group gets one chocolate kiss for each stamp earned.

5 min Closure Students:

o Write down notes template in their logs/planners o Write down practice sheet and answer key in their logs/planners o Write down homework in their logs/planners

DATE:

Calculus Section: Name: Unit 1, Lesson 2: Warm-Up 1) Determine if each graph represents a function. 2) Find the domain and range of each graph.

DATE:

Calculus Section: Name: Unit 1, Lesson 2: Lecture Notes

Main Concept

Evaluating Functions

Evaluating Composite Functions

f g

f ( g(9) ) = f ( g(15) ) =

f ( g(−4) ) = g( f (−1) ) =

x y -4 8 9 -1 15 0 2 6

f ( g(2) ) =

g( f (−5) ) =

8 0.5 -12 6 1 4

3 -2 10 0

f 10 − 2( ) =

f 16( ) =

f 7

14

⎛

⎝⎜⎞

⎠⎟=

6

-1

2

0

-4

7

5

DATE:

f ( g(2) ) =

g( f (−3) ) =

You try:

f ( g(−1) ) =

f ( f (−4) ) =

f (x ) = x 2 − 2x + 3 g(x ) = 4x − 2

f ( g(x ) ) = g( f (x ) ) =

Is f ( g(x ) ) = g( f (x ) ) ?

Watch out for constants!

f (x ) = x 2 − 2x + 3 g(x ) = 8

f ( g(x ) ) = g( f (x ) ) =

Relay Race Progress Checker

Group Members:

Round Completed! Round Completed!

1 4

2 5

3 6

Relay Race Progress Checker

Group Members:

Round Completed! Round Completed!

1 4

2 5

3 6

Round 1: f (x ) = −2x + 3 g(x ) = x 2 −1 h(x ) = 7

f ( g(x ) ) = g( f (x ) ) =

g( h(x ) ) = h( g(x ) ) =

Round 1: f (x ) = −2x + 3 g(x ) = x 2 −1 h(x ) = 7

f ( g(x ) ) = g( f (x ) ) =

g( h(x ) ) = h( g(x ) ) =

Round 2: f (x ) = x 2 − 3x −1 g(x ) = 5x + 2 h(x ) = 3x

f ( g(x ) ) = g( f (x ) ) =

f ( h(x ) ) = h( f (x ) ) =

Round 2: f (x ) = x 2 − 3x −1 g(x ) = 5x + 2 h(x ) = 3x

f ( g(x ) ) = g( f (x ) ) =

f ( h(x ) ) = h( f (x ) ) =

Round 3: f (x ) = x3 g(x ) = 3− x h(x ) = 3− 4x 2

g( g(x ) ) = h( f (x ) ) =

h( g(x ) ) = g( h(x ) ) =

Round 3: f (x ) = x3 g(x ) = 3− x h(x ) = 3− 4x 2

g( g(x ) ) = h( f (x ) ) =

h( g(x ) ) = g( h(x ) ) =

Round 4: f (x ) = −5 g(x ) = −2x 2 − x h(x ) =10 − 3x

g( f (x ) ) = f ( g(x ) ) =

h( h( f (x ) ) ) = g( h(x ) ) =

Round 4: f (x ) = −5 g(x ) = −2x 2 − x h(x ) =10 − 3x

g( f (x ) ) = f ( g(x ) ) =

h( h( f (x ) ) ) = g( h(x ) ) =

Round 5: f (x ) =

14

x + 2 g(x ) = x 2 +

12

x −1 h(x ) =

34

f ( h(x ) ) = h( f (x ) ) =

g( h(x ) ) = f ( g(x ) ) =

Round 5: f (x ) =

14

x + 2 g(x ) = x 2 +

12

x −1 h(x ) =

34

f ( h(x ) ) = h( f (x ) ) =

g( h(x ) ) = f ( g(x ) ) =

Round 6: f (x ) = 6 + 0.4x g(x ) = 0.1x 2 − 3.5x + 2 h(x ) = 0.9x

h( g(x ) ) = g( h(x ) ) =

f ( g(x ) ) = g( f (x ) ) =

Round 6: f (x ) = 6 + 0.4x g(x ) = 0.1x 2 − 3.5x + 2 h(x ) = 0.9x

h( g(x ) ) = g( h(x ) ) =

f ( g(x ) ) = g( f (x ) ) =

Calculus Section: Name: Homework #1-2

Composite Functions

f (x ) = 4 − 2x g(x ) = 2x 2 − 3x + 5 h(x ) = 3x − 2 x −10

1. f ( h(6) ) = 2. h( g(0) ) = 3. f ( f (−5) ) = 4. g( f (1) ) = 5.

f ( h( g(3) ) )=

You have a skills test next class. Make sure to prepare!

1) #1-1 part 1 (Do the attached problems. Be ready!)

DATE:

D

ete

rmin

e i

f it

’s a

fu

nct

ion

(g

rap

hic

all

y)

1. Is each relation a function? Write yes or no below each one.

De

term

ine

if

it’s

a f

un

ctio

n

(tab

les,

arr

ow

map

s)

2. Is each relation a function? Write yes or no below each one.

a) b) x 8 3 4 -2 8 7

y 6 3 0 -5 -1 8

De

term

ine

d

om

ain

an

d r

an

ge

(tab

les,

arr

ow

m

ap

s)

3. Write the domain and range for the relations in problem 2.

a) Domain: b) Domain: Range: Range:

De

term

ine

do

main

an

d r

an

ge

(gra

ph

icall

y)

4. Write the domain and range for the graph of f(x) below.

Domain: Range:

x

y

x

y

x

y

x

y

°

•

5 -2 3 ½

1 0 2

f(x)

•

•

Evalu

ate

fu

nct

ion

s (a

lge

bra

icall

y)

5. Given that f(x) = 2x2 + 5x – 4, find f(-3).

Evalu

ate

fu

nct

ion

s (g

rap

hic

all

y)

6. Use the graph of f(x) to evaluate

each:

a) f(-4) = b) f(4) =

c) f(-51) =

d) f(0) =

Evalu

ate

fu

nct

ion

s (w

ith

tab

les)

7. Use the table of values to evaluate each:

a) f(0) =

b) g(1) =

x 5 -5 3 8 0 1 9 y 12 10 -1 3 -5 0 7

x -3 2 8 1 -4 7 6 y 0 -5 -5 8 1 3 5

f

g