function composition and operations activity new from old 1

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Unit 1 • Linear Systems and Matrices 33

My Notes

ACTIVITY

1.4Function Composition and OperationsNew from OldSUGGESTED LEARNING STRATEGIES: Marking the Text, Create Representations, Quickwrite, Identify a Subtask, Group Presentation

Jim Green has a lawn service called Green’s Grass Guaranteed. On every job, Jim charges a fi xed $30 fee to cover equipment and travel expenses plus a $20 per hour labor charge.

1. On a recent job, Jim worked for six hours. What was the total charge for this job?

2. If Jim works for T hours, what will he charge for the job? Write your answer as a cost function F, where F(T ), or C, is Jim’s charge for T hours of work.

It takes Jim four hours to mow one acre. Jim prepares a cost estimate for each customer based on the size (number of acres) of the property.

3. Th e APCON company is one of Jim’s customers. APCON has two acres that need mowing. How many hours does that job take?

4. Another customer has A acres of property. Write an equation in terms of A for the number of hours T it will take Jim to mow the property.

5. How much will Jim charge APCON to mow its property? Justify your answer.

Th e functions in Items 2 and 4 relate three quantities that vary, based on the needs of Jim’s customers: • Th e size in acres A of the property • Th e time in hours T needed to perform the work • Th e cost in dollars C of doing the work.

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34 SpringBoard® Mathematics with Meaning™ Algebra 2

My Notes

Function Composition and Operations ACTIVITY 1.4continued New from OldNew from Old

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Graphic Organizer

6. Complete the table below by writing the rate of change with units and fi nding the slope of the graph of the function.

Function Rate of Change (with units) SlopeF(T ): C =G(A): T =

7. Complete the table below by naming the measurement units for the domain and range of each function.

Function Notation

Description of Function

Domain(units)

Range(units)

F(T) cost for jobG(A) time to mow

8. Calculating the cost to mow a lawn is a two-step process. Complete the graphic organizer below by describing the input and output, including units, for each part of the process.

Input:

Time to Mow

Output:

Output:

Input:

Cost for Job

Input:

Time to Mow

Output:

Output:

Input:

Cost for Job

In a linear function f (x) = mx + b, the y-intercept is b. Thevariable m is the rate of change in the values of the function-the change of units of f (x) per change of unit of x. When the function is graphed, the rate of change is interpreted as the slope. Soy = mx + b is called theslope-intercept form of a linear equation.


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My Notes

ACTIVITY 1.4continued

Function Composition and Operations New from OldNew from Old

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Summarize/Paraphrase/Retell, Think/Pair/Share, Graphic Organizer, Marking the Text

Th e graphic organizer shows an operation on two functions, called a composition. Th e function that results from using the output of the fi rst function as the input for the second function is a composite function.

In this context, the composite function is formed by the time-to-mow function and the cost-for-job function. Its domain is the input for the time function, and its range is the output from the cost function.

9. Th e cost to mow is a composite function. Describe its input and output as you did in Item 8.

When a composite function is formed, the function is oft en named to show the functions used to create it. Th e cost-to-mow function, F(G(A)), is composed of the cost-for-job and the time-to-mow functions.

Th e F(G(A)) notation implies that A was assigned a value G(A) by the time-to-mow function. Th en the resulting G(A) value was assigned a value F(G(A)) by the cost-to-mow function.

10. Complete the table by writing a description for the composite function F(G(A)). Th en name the measurement units of the domain and range.

Function Notation Description of Function

F(G(A))

Domain (units) Range (units)

Output:

Input:

Time to Mow

Cost for Job

Input:

Number of Hours

Output:

Input:

Time to Mow

Cost for Job

Input:

Number of Hours

ACADEMIC VOCABULARY

In a composite function, the range of the fi rst function becomes the domain for the second function.


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My Notes


SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation

Jim wants to write one cost function for mowing A acres of property. To write the cost C as a function of A acres of property, he substitutes T into the functions and simplifi es.

F(T ) = F(G(A)) Substitute G(A) for T in the cost function.

F(G(A)) = F(4A) G(A) = 4A, so write the function in terms of A.

= 30 + 20(4A) Substitute 4A into the original F(T ) function.

F(G(A)) = 30 + 80A

11. Write a sentence to explain what the expression F(G(2)) represents. Include appropriate units in your explanation.

12. Why might Jim want a single function to determine the cost of a job when he knows the total number of acres?

13. Write a sentence to explain what the expression F(G(50)) represents. Include appropriate units in your explanation.

14. Write a sentence to explain what information the equationF(G(A)) = 50 represents. Include appropriate units in yourexplanation.


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My Notes



SUGGESTED LEARNING STRATEGIES: Note Taking, Create Representations

A composition of functions forms a new function by substituting the output of the inner function into the outer function. Th e function y = f ( g (x)) is a composition of f and g where g is the inner function and f is the outer function.

15. Use the tables of values below to evaluate each expression.

x f(x)1 32 23 14 4

x g(x)1 42 33 24 1

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

c. g ( f (3)) d. f ( g (3))

16. Using f and g from Item 15, 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)) 1234

b. x ( g � f )(x) = g ( f (x)) 1234

WRITING MATH

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

( f � g)(x) = f ( g(x))

Read the notation as “f of g of x.”

The order matters when you compose two functions.

y = g( f (x)) andy = f ( g(x)) are twodifferent functions.


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My Notes


SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Group Presentation

For Items 17–21, use these three functions:

• f (x) = x2

• g(x) = 2x - 1 • h(x) = 4x - 3

17. Evaluate each expression.

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

18. Write each composite function in terms of x.

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

19. Verify that you composed g and f correctly by evaluating g( f (2)) and f ( g(2)) using the functions you wrote in Item 18. Compare your answers with those from Item 17.

CONNECT TO APAP

In AP Calculus, you will identify the “inner” function and the “outer” functions 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) = x 2 .

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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Group Presentation, Activating Prior Knowledge

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

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

21. a. Evaluate g ( g (2)).

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

Addition, subtraction, multiplication and division are operations on real numbers. You can also perform those operations with functions.

22. Given g(x) = 2x - 1 and f (x) = 3x + 2, fi nd each function.

a. ( g + f )(x)

b. ( g - f )(x)


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My Notes


SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge

22. (continued)

c. ( g · f )(x)

d. ( g ÷ f )(x), f (x) ≠ 0

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

Hannah’s Housekeeping charges a $20 fl at fee plus $12 an hour to clean a house.

1. What would Hannah charge if it took four hours to clean a house?

2. Write a function F(H) for the cost to clean a house for H hours.

3. What are the units of the domain and range of this function?

4. What is the slope of this function? Interpret the slope as a rate of change.

Hannah’s Housekeeping can clean one room every half hour.

5. How long does it take Hannah to clean six rooms?

6. Write a function G(R) for the hours needed to clean R rooms.

7. How much would Hannah charge to clean a 10-room home?

8. Write a function F(G(R)) to represent the cost of cleaning R rooms.

9. What is the value and meaning of F(G(12))?

For Items 10–14, use the following functions:

• f (x) = 5x + 1

• g (x) = 3x - 4

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

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

12. Draw a diagram to show the composition of g (x) and f (x).

13. Write the functions ( f + g)(x) and ( f - g)(x).

14. Write the functions ( f · g)(x) and ( f ÷ g)(x), g (x) ≠ 0.

15. MATHEMATICAL R E F L E C T I O N

Explain in detail how a composition of functions

forms a new function from the old (original) functions.

Be sure not to confuse ( g � f )(x), composition, and ( g · f )(x), multiplication. The fi rst is the composition of the two functions and the second is the product of the two functions.


function composition and operations activity new from old 1

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