## Engage NY Eureka Math Precalculus Module 3 Lesson 16 Answer Key

### Eureka Math Precalculus Module 3 Lesson 16 Example Answer Key

Example 1.

Consider the tables from the opening scenario

a. Do the tables appear to represent functions? If so, define the function represented in each table using a verbal description.

Answer:

Both tables appear to represent functions because for each input in the domain, there is exactly one output.

In the first table, the depth of the diver is a function of the time spent descending.

In the second table, the pressure on the diver is a function of the diver’s depth.

b. What are the domain and range of the functions?

Answer:

For the first table, the domain and range are nonnegative real numbers.

For the second table, the domain is nonnegative real numbers, and the range is real numbers greater than or equal to 1.

c. Let’s define the function in the first table as d = f(s) and the function in the second table as p = g(d).

Use function notation to represent each output, and use the appropriate table to find its value.

i. Depth of the diver at 80 seconds

Answer:

d = f(80) = 65. After 80 seconds, the diver has descended 65 meters.

ii. Pressure on the diver at a depth of 60 meters

Answer:

p = g(60) = 7. At a depth of 60 meters, there are 7 atmospheres of pressure on the diver.

d. Explain how we could determine the pressure applied to a diver after 120 seconds of descent.

Answer:

We could use the first table to determine the depth that corresponds to a descent time of 120 seconds and then use the second table to find the pressure that corresponds to this depth.

e. Use function notation to represent part (d), and use the tables to evaluate the function.

Answer:

g(f(120)) = g(90) = 10

f. Describe the output from part (e) in context.

Answer:

The pressure applied to a diver 120 seconds into a descent is 10 atmospheres.

Example 2.

Consider these functions:

f: Animals → Counting numbers

Assign to each animal the number of legs it has.

g: People → Animals

Assign to each person his favorite animal.

Determine which composite functions are defined. If defined, describe the action of each composite function.

a. f ∘ g

Answer:

Assign a person to her favorite animal, then assign the animal to its number of legs: The composite function is defined, and assign each person to the number of legs of her favorite animal

b. f ∘ f

Answer:

This composition is not defined. Function f assigns a number to an animal, but it cannot accept the number that it outputs as an input. The range of f is not contained within the domain of f.

c. g ∘ f

Answer:

This composition is not defined. Function f assigns each animal to a number. Function g accepts only people as inputs, so it cannot accept the number output by function f. The range of f is not contained within the domain of g.

d. f ∘ g ∘ g

Answer:

The composition g ∘ g is not defined. Function g assigns each person to an animal, but it cannot accept the animal that it outputs as an input. The range of g is not contained within the domain of g.

### Eureka Math Precalculus Module 3 Lesson 16 Exercise Answer Key

Exercises

Exercise 1.

Let f(x) = x^{2} and g(x) = x + 5. Write an expression that represents each composition:

a. g(f(4))

Answer:

g(f(4)) = g(4^{2} ) = g(16) = (16 + 5) = 21

b. f(g(4))

Answer:

f(g(4)) = f(4 + 5) = f(9) = 9^{2} = 81

c. (f ∘ g)(x)

Answer:

(f ∘ g)(x) = f(x + 5) = (x + 5)^{2} = x^{2} + 10x + 25

d. (f ∘ g)(√(x + 5))

Answer:

(f ∘ g)(\(\sqrt{x + 5}\)) = f(\(\sqrt{x + 5}\) + 5) = (\(\sqrt{x + 5}\) + 5)^{2} = x + 10\(\sqrt{x + 5}\) + 30

Exercise 2.

Suppose a sports medicine specialist is investigating the atmospheric pressure placed on competitive free divers during their descent. The following table shows the depth, d, in meters of a free diver s seconds into his descent. The depth of the diver is a function of the number of seconds the free diver has descended, d = f(s).

The pressure, in atmospheres, felt on a free diver, d, is a function of his depth, p = g(d).

a. How can the researcher use function composition to examine the relationship between the time a diver spends descending and the pressure he experiences? Use function notation to explain your response.

Answer:

The function g(f(s)) represents the pressure experienced by a diver who has been descending for s seconds. The function f assigns a depth in meters to each time s, and the function g assigns a pressure to each depth. Then g ∘ f assigns a pressure to each time.

b. Explain the meaning of g(f(0)) in context.

Answer:

The value of g(f(0)) is the pressure on a free diver at his depth 0 seconds into the descent.

c. Use the charts to approximate these values, if possible. Explain your answers in context.

i. g(f(70))

Answer:

5.5 atmospheres; this is the pressure on the free diver at his depth 70 seconds into his dive (d = 55 meters).

ii. g(f(160))

Answer:

Approximately 13 atmospheres; this is the pressure on the free diver at his depth 160 seconds into his dive (d = 130 meters).

### Eureka Math Precalculus Module 3 Lesson 16 Problem Set Answer Key

Question 1.

Determine whether each rule described represents a function. If the rule represents a function, write the rule using function notation, and describe the domain and range.

a. Assign to each person her age in years.

Answer:

Yes. f: People → Numbers

Domain: set of all living people. Range: {0,1,2,3,…,130}

b. Assign to each person his height in centimeters.

Answer:

Yes. f: People → Numbers

Domain: set of all living people. Range: {50,51,…280}

c. Assign to each piece of merchandise in a store a bar code.

Answer:

Yes. f: Products → Bar codes

Domain: each piece of merchandise in the store. Range: {unique bar codes}

d. Assign each deli customer a numbered ticket.

Answer:

Yes. f: People → Numbered tickets

Domain: set of people that are waiting in the deli. Range: {Numbered tickets}

e. Assign a woman to her child.

Answer:

No. There are many women who have more than one child and many who have no children.

f. Assign to each number its first digit.

Answer:

Yes. f: Counting numbers → Counting numbers

Domain: set of all counting numbers. Range: {1,2,3,4,5,6,7,8,9}

g. Assign each person to the city where he was born.

Answer:

Yes. f: People → Cities

Domain: set of all counting numbers. Range: {Cities} .

Question 2.

Let L: Animal → Counting numbers

Assign each animal to its number of legs.

F: People → Animals

Assign each person to his favorite animal.

N: People → Alphabet

Assign each person to the first letter of her name.

A: Alphabet → Counting numbers

Assign each letter to the corresponding number 1–26.

S: Counting Numbers → Counting numbers

Assign each number to its square.

Which of the following compositions are defined? For those that are, describe the effect of the composite function.

a. L ∘ F

Answer:

The composite function is defined. The function assigns each person to his favorite animal and then to the number of legs the animal has.

b. N ∘ L

Answer:

This composition is not defined. The function L assigns each animal to its number of legs, but function N accepts only people as inputs. The range of L is not contained in the domain of N.

c. A ∘ L

Answer:

This composition is not defined. The function L assigns each animal to its number of legs, but function A accepts only letters as inputs. The range of L is not contained in the domain of A.

d. A ∘ N

Answer:

This composite function is defined. The function assigns a person to the first letter of his name, and then to the number 1–26 that corresponds to that letter.

e. N ∘ A

Answer:

This composition is not defined. The function A assigns each letter of the alphabet to its corresponding number, but function N accepts only people as inputs. The range of A is not contained in the domain of N.

f. F ∘ L

Answer:

This composition is not defined. The function L assigns each animal to its number of legs , but function F accepts only people as inputs. The range of L is not contained in the domain of F.

g. S ∘ L ∘ F

Answer:

This composition is defined. The function assigns a person to her favorite animal, then to the number of legs of that animal, and then to the square of that number.

h. A ∘ A ∘ N

Answer:

This composition is not defined. The composition A ∘ N is defined, and outputs a number, which is not a valid input to function A. The range of A ∘ N is not contained within the domain of A.

Question 3.

Let f(x) = x^{2} – x, g(x) = 1 – x.

a. f ∘ g

Answer:

f(g(x)) = f(1 – x) = 1 – 2x + x^{2} – 1 + x = x^{2} – x

b. g ∘ f

Answer:

g(f(x)) = g(x^{2} – x) = 1 – x^{2} + x = – x^{2} + x – 1

c. g ∘ g

Answer:

g(g(x)) = g(1 – x) = 1 – 1 + x = x

d. f ∘ f

Answer:

f(f(x)) = f(x^{2} – x) = x^{4} – 2x^{3} + x^{2} – x^{2} + x = x^{4} – 2x^{3} + x

e. f(g(2))

Answer:

f(g(2)) = f( – 1) = 2

f. g(f( – 1))

Answer:

g(f( – 1)) = g(2) = – 1

Question 4.

Let f(x) = x^{2}, g(x) = x + 3.

a. g(f(5))

Answer:

g(f(5)) = g(5^{2}) = g(25) = 25 + 3 = 28

b. f(g(5))

Answer:

f(g(5)) = f(5 + 3) = f(8) = 8^{2} = 64

c. f(g(x))

Answer:

f(g(x)) = f(x + 3) = (x + 3)^{2} = x^{2} + 6x + 9

d. g(f(x))

Answer:

g(f(x)) = g(x^{2}) = x^{2} + 3

e. g(f(\(\sqrt{x + 3}\)))

Answer:

g(f(\(\sqrt{x + 3}\))) = g(x + 3) = x + 6

Question 5.

Let f(x) = x^{3}, g(x) = \(\sqrt [ 3 ]{ x }\).

a. f ∘ g

Answer:

f(g(x)) = f(\(\sqrt [ 3 ]{ x }\)) = x

b. g ∘ f

Answer:

g(f(x)) = g(x^{3} ) = x

c. f(g(8))

Answer:

f(g(8)) = f(2) = 8

d. g(f(2))

Answer:

g(f(2)) = g(8) = 2

e. f(g( – 8))

Answer:

f(g( – 8)) = f( – 2) = – 8

f. g(f( – 2))

Answer:

g(f( – 2)) = g( – 8) = – 2

Question 6.

Let f(x) = x^{2}, g(x) = \(\sqrt{x + 3}\)

a. Show that (f(x + 3)) = |x + 3| + 3.

g(f(x + 3)) = g((x + 3)^{2} ) = \(\sqrt{(x + 3)^{2}}\) + 3 = |x + 3| + 3

b. Does (x) = |x + 3| + 3 = (x) = |x| + 6? Graph them on the same coordinate plane.

Answer:

No, they are not equal.

For |x + 3| + 3, if x + 3≥0, |x + 3| + 3 = x + 6; if x + 3<0, |x + 3| + 3 = – x.

For |x| + 6, if x≥0, |x| + 6 = x + 6; if x<0, |x| + 6 = – x + 6.

Question 7.

Given the chart below, find the following:

a. f(g(0))

Answer:

2

b. g(k(2))

Answer:

4

c. k(g( – 6))

Answer:

0

d. g(h(4))

Answer:

2

e. g(k(4))

Answer:

g(3) is not defined.

f. (f ∘ g ∘ h)(2)

Answer:

– 6

g. (f ∘ f ∘ f)(0)

Answer:

2

h. (f ∘ g ∘ h ∘ g)(2)

Answer:

2

Question 8.

Suppose a flu virus is spreading in a community. The following table shows the number of people, n, who have the virus d days after the initial outbreak. The number of people who have the virus is a function of the number of days, n = f(d).

There is only one pharmacy in the community. As the number of people who have the virus increases, the number of boxes of cough drops, b, sold also increases. The number of boxes of cough drops sold on a given day is a function of the number of people who have the virus, b = g(n), on that day.

a. Find g(f(1)), and state the meaning of the value in the context of the flu epidemic. Include units in your answer.

Answer:

Because f(1) = 4 and g(f(1)) = 14, on day one, there were four people infected, and there were fourteen boxes of cough drops sold at the pharmacy.

b. Fill in the chart below using the fact that b = g(f(d)).

Answer:

c. For each of the following expressions, interpret its meaning in the context of the problem, and if possible, give an approximation of its value.

a. g(f(4))

Answer:

g(f(4)) = g(14) = 22

On the fourth day of the outbreak, 22 boxes of cough drops were sold.

b. g(f(16))

Answer:

g(f(16)) = g(50) = 102

On the sixteenth day of the outbreak, 102 boxes of cough drops were sold.

c. f(g(9))

Answer:

We can compute f(g(9)) = f(16) = 50 but it does not make sense in the context of this problem. The output g(9) = 16 represents the number of boxes of cough drops sold when 9 people are infected. The output f(16) = 50 represents the number of people infected on day 16 of the outbreak. However, f is a function of days, not a function of the number of boxes of cough drops sold.

### Eureka Math Precalculus Module 3 Lesson 16 Exit Ticket Answer Key

Question 1.

Let f(x) = x^{2} and g(x) = 2x + 3. Write an expression that represents each composition:

a. (g ∘ f)(x)

Answer:

(g ∘ f)(x) = g(x^{2} ) = 2(x^{2} ) + 3 = 2x^{2} + 3

b. f(f( – 2))

Answer:

f(f( – 2)) = f(4) = 4^{2} = 16

c. (f ∘ g)(1/x)

Answer:

(f ∘ g)(\(\frac{1}{x}\)) = f(2(\(\frac{1}{x}\)) + 3) = (\(\frac{2}{x}\) + 3)^{2} = \(\frac{4}{x^{2}}\) + \(\frac{12}{x}\) + 9

Question 2.

A consumer advocacy company conducted a study to research the pricing of fruits and vegetables. They collected data on the size and price of produce items, including navel oranges. They found that, for a given chain of stores, the price of oranges was a function of the weight of the oranges, p = f(w).

The company also determined that the weight of the oranges measured was a function of the radius of the oranges, w = g(r).

a. How can the researcher use function composition to examine the relationship between the radius of an orange and its price? Use function notation to explain your response.

Answer:

The function f(g(r)) represents the price of oranges as a function of the radius of the oranges.

b. Use the table to evaluate f(g(2)), and interpret this value in context.

Answer:

f(g(2)) = f(0.5) = 0.65

The price of oranges with a radius of 2 inches is $0.65.