More Function Composition - If students need additional help with understanding the meaning of

If students need additional help with understanding the meaning of composite function notation, a mini-lesson is available to provide practice

Lesson 5-3 More Function Composition

Learning Targets:

•

Write the composition of two functions.

•

Evaluate the composition of two functions.

SUGGESTED LEARNING STRATEGIES: Note Taking, Create Representations, Think-Pair-Share, Group Presentation, Debriefing A composition of functions forms a new function by substituting the output of the inner function into the outer function. The function y = f (g(x)) is a composition of f and g where g is the inner function and f is the outer function.

1. The tables show information about Jim’s mowing service. Use the tables to evaluate each expression. Then tell what the expression represents.

Area of Property a

(acres)

Time to Mow

t(a) (hours) Time to Mow

t (hours) Cost to Mow c(t) ($)

1 4 4 110

2 8 8 190

3 12 12 270

4 16 16 350

a. t(4) b. c(4) c. c(t(1)) d. c(t(4))

2. Reason quantitatively. Use the tables of values below to evaluate each expression.

x f(x) x g(x)

1 3 1 4

2 2 2 3

3 1 3 2

4 4 4 1

a. f (3) b. g(3)

c. g ( f (3)) d. f (g(3)) The order matters when you

compose two functions.

y = g(f (x)) and y = f (g(x)) are two different functions.

MATH TIP

16; It takes 16 hours to mow 4 acres.

110; It costs $110 for 4 hours of mowing.

110; It costs $110 to mow 1 acre.

350; It costs $350 to mow 4 acres.

1 4

2 2

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ACTIVITY 5

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ACTIVITY 5

^Continued

Lesson 5-3 PLAN

Pacing: 1 class period Chunking the Lesson

#1 #2–3

Check Your Understanding

#7–11

Check Your Understanding Lesson Practice

TEACH

Bell-Ringer Activity

Have students review (without function notation) substituting values from one expression into another and simplifying, in order to help them with the

composite functions.

1. Find 2x + 9, if x = 14. [37]

2. Find 3r − 4 in terms of x, if r = x − 6. [3x − 22]

3. Find −4a in terms of b, if a = b + 12. [−4b − 48]

Have students discuss their results prior to moving forward with the lesson.

T

EACHER

to T

EACHER

The main focus of the remainder of this activity is practice with composition of functions. Students compose functions numerically and algebraically. They also learn an alternate notation for composition and work with more typical

representations of functions using f, g, and h as functions of x.

1 Create Representations In Item 1, the use of the two tables, one for each function, provides a visual way of demonstrating how the output, or range, of the first (inner) function becomes the domain of the second (outer) function, as column 2 from the first table is mirrored as column 1 in the second table. The first table represents the t function, and the second table represents the c function.

My Notes

3. Using f and g from Item 2, complete each table of values to represent the composite functions ( f g)(x) and ( g f )(x).

a. x ( f g)(x) = f(g(x))

1 4

2 ¹

3 ²

4 3

Lesson 5-3

More Function Composition

b. x (g f )(x) = g(f(x))

1 2

2 ³

3 ⁴

4 1

4. What is does the notation ( g h)(t) represent? What is another way you can write ( g h)(t)?

5. Reason abstractly. Explain how (f g)(x) is different from (f

⋅

^g)(x).

6. Given that p(t) = t² + 4 and q(t) = t + 3, write the equation for ( p q)(t). Explain how you determined your answer.

Check Your Understanding

For Items 7–11, use these three functions:

•

^{f (x) = x}²

•

g (x) = 2x − 1

•

h(x) = 4x − 3

7. Evaluate each expression.

a. g ( f (2)) b. f (g (2))

8. Write each composite function in terms of x.

a. y = g ( f (x)) b. y = f ( g (x))

9. Verify that you composed g and f correctly by evaluating g ( f (2)) and f ( g (2)) using the functions you wrote in Item 8. Compare your answers with those from Item 7.

The notation (f g)(x) represents a composition of two functions.

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

Read the notation as “f of g of x.”

WRITING MATH

In AP Calculus, you will identify the

“inner” function and the “outer”

function that form a composite function.

For example, the function h(x) = f (g(x)) = (2x + 3)² could be the composition of the inner function g(x) = 2x + 3 and the outer function f (x) = x².

AP CONNECT TO

g(f(x)) = 2x² − 1 f(g(x)) = (2x − 1)²

= 4x² − 4x + 1

g(f(2)) = 2(2)² − 1 = 7 f(g(2)) = (2(2) − 1)² = 9

Activity 5 • Function Composition and Operations 85 continued ACTIVITY 5

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AP TO CONNECT

ACTIVITY 5

^Continued

2–3 Create Representations, Predict and Confirm, Debriefing Give students an opportunity to work through Item 2 on their own. If students need help, demonstrate each step in the process of composing the functions numerically.

The composite functions that students write in Item 3 will be numeric only, within a table of values. Students are not expected to represent either function or the composite functions algebraically.

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to function compositions.

Answers

4. Sample answer: (g o h)(t) represents a composition of the functions g and h in which h is the inner function and g is the outer function.

(g o h)(t) can also be written as g(h(t)).

5. (f o g)(x) is a composition of the functions f and g: (f o g)(x) = f(g(x)).

By contrast, (f

⋅

g)(x) is the product of the functions f and g: (f

⋅

^{g)(x) =}

f(x)

⋅

^g(x).

6. (p o q)(t) = (t + 3)² + 4 =

t² + 6t + 13; Sample explanation: To find the rule for (p o q)(t), I replaced t in the rule for p(t) with the rule for q(t).

7–11 Think-Pair-Share, Create Representations, Group

Presentation, Debriefing These items provide an opportunity for students to compose two functions algebraically.

Most of these items use only linear or simple quadratic functions. You may find it necessary to provide further practice with composing different types of functions as they are introduced later in the year.

Students should also gain familiarity with the composition notation f o g and g o f.

In order to decompose a function, students will need to recognize the inner and outer parts of the composite function in order to correctly apply the chain rule when they study calculus.

Examples:

Composite

f(g(x)) = 5x−3 Inner Function

g(x) = 5x − 3 Outer Function f x( ) (= x) Composite

f(g(x)) = 4 x ² − 1 Inner Function

g(x) = x ² Outer Function f(x) = 4x − 1

My Notes

Lesson 5-3 More Function Composition

10. a. Evaluate h( g (3)).

b. Write the composition (h g)(x) in terms of x.

11. a. Evaluate g ( g (2)).

b. Write the composition ( g g)(x) in terms of x.

Check Your Understanding

12. Explain how you found the rule for the composition ( g g)(x) in Item 11b.

13. Given that p(n) = 4n and q(n) = n + 2, for what value of n is (p q)(n) = 8? Explain how you determined your answer.

LESSON 5-3 PRACTICE

For Items 14 and 15, use the following functions:

•

f (x) = 5x + 1

•

g (x) = 3x − 4

14. Evaluate f (2), g(2), ( f g)(2), and ( g f )(2).

15. Write the composite functions h(x) = g ( f (x)) and k(x) = f ( g (x)).

The jeans at a store are on sale for 20% off, and the sales tax rate is 8%. Use this information for Items 16–18.

16. Write a function s(p) that gives the sale price of a pair of jeans regularly priced at p dollars.

17. Write a function t(p) that gives the total cost including tax for a pair of jeans priced at p dollars.

18. Construct viable arguments. A customer wants to buy a pair of jeans regularly priced at $25. Does it matter whether the sales clerk applies the sale discount first or adds on the sales tax first to find the total cost? Use compositions of the functions s and t to support your answer.

As shown in Item 11, the inner and outer functions that form a composite function can be the same function.

MATH TIP

h(g(3)) = h(2(3) − 1) = h(5) = 4(5) − 3 = 17

h(g(x)) = h(2x − 1) = 4(2x − 1) − 3 = 8x − 4 − 3 = 8x − 7

5; g(g(2)) = g(2(2) − 1) = g(3) = 2(3) − 1 = 5

(g g)(x) = g(2x − 1) = 2(2x − 1) − 1 = 4x − 3

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ACTIVITY 5

^Continued

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to evaluating the composition of functions.

Answers

12. Sample answer: I took the rule for g, 2x − 1, and substituted it for x in the rule for g: 2(2x − 1) − 1. Then I simplified the resulting expression:

2(2x − 1) −1 = 4x − 3.

13. n = 0; Sample explanation: First I wrote the rule for (p o q)(n):

(p o q)(n) = 4(n + 2) = 4n + 8.

Then I substituted 8 for (p o q)(n) and solved for n.

(p o q)(n) = 4n + 8

8 = 4n + 8

0 = 4n

0 = n

ASSESS

Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning.

See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity.

LESSON 5-3 PRACTICE

14. f(2) = 11; g(2) = 2; f(g(2)) = 11;

g(f(2)) = 29

15. h(x) = g(f(x)) = 3(5x + 1) − 4 = 15x − 1; k(x) = f(g(x)) = 5(3x − 4) + 1 = 15x − 19 16. s(p) = 0.8p

17. t(p) = 1.08p

18. No. Sample explanation: The composite function s(t(p)) = 0.8(1.08p) = 0.864p gives the total cost of the jeans if the sales tax is added on before the discount is applied. The composite function t(s(p)) = 1.08(0.8p) = 0.864p gives the total cost of the jeans if the discount is applied before the sales tax is added on. Both composite functions have the same rule, so both give the same total cost for the jeans: 0.864($25) = $21.60.

ADAPT

Check students’ answers to the Lesson Practice to ensure that they understand how to write and evaluate function compositions. Students who make errors in evaluation may be evaluating the functions in the wrong order. Have them write the composition without using the notation o, if necessary, and remind them to begin in the innermost parentheses and work outward.

Function Composition and Operations New from Old

ACTIVITY 5 PRACTICE

In document Springboard Unit 1 - Equations, Inequalities and Functions (Page 86-89)