## Engage NY Eureka Math Algebra 1 Module 3 Lesson 9 Answer Key

### Eureka Math Algebra 1 Module 3 Lesson 9 Example Answer Key

Example 1.
Define the Opening Exercise using function notation. State the domain and the range.
f: {animals pictured} → {words listed}
Assign each animal to its proper name.
Domain: {the four animal pictures}
Range: {elephant,camel,polar bear,zebra}

Example 2.
Is the assignment of students to English teachers an example of a function? If yes, define it using function notation, and state the domain and the range.
Assign students in this class to the English teacher according to their class schedule.
Domain: the students in this class
Range: the English teachers in the school whose students are in your class

Example 3.
Let X = {1, 2, 3, 4} and Y = {5, 6, 7, 8, 9}. f and g are defined below.
f:X → Y g:X → Y
f = {(1, 7),(2, 5),(3, 6),(4, 7)} g = {(1, 5),(2, 6),(1, 8),(2, 9),(3, 7)}
Is f a function? If yes, what is the domain, and what is the range? If no, explain why f is not a function.
Yes, f is a function because each element of the domain is matched to exactly one element of the range.
The domain is {1, 2, 3, 4} and the range is {5, 6, 7}.

Is g a function? If yes, what is the domain and range? If no, explain why g is not a function.
No, g is not a function because an element of the domain is assigned to more than one element of the range.
For example, the 1 is matched to both 5 and 8.

What is f(2)?
f(2) = 5 since 2 is matched to 5.

If f(x) = 7, then what might x be?
If f(x) = 7, then x = 1 or x = 4.

### Eureka Math Algebra 1 Module 3 Lesson 9 Exercise Answer Key

Opening Exercise
Match each picture to the correct word by drawing an arrow from the word to the picture.

Introduce that there is a correspondence between the pictures and the words that name them based on a joint understanding of animals. However, consider pointing out that while students might have naturally matched the pictures with the words, there are certainly other ways to pair the pictures and words. For example, match the pictures to the words left to right and then top to bottom. According to this convention, the polar bear picture would be assigned to the word elephant.

Even a simple example like this begins to build the concept of function in students’ minds that there must be a set of inputs (domain) paired with a set of outputs (range) according to some criteria. As these lessons develop, introduce algebraic functions, and students learn that by substituting every value in the domain of a function into all instances of the variable in an algebraic expression and evaluating that expression, they can determine the f(x) value associated with each x in the domain.

FUNCTION: A function is a correspondence between two sets, X and Y, in which each element of X is matched to one and only one element of Y. The set X is called the domain of the function.

The notation f:X → Y is used to name the function and describes both X and Y. If x is an element in the domain X of a function f:X → Y, then x is matched to an element of Y called f(x). We say f(x) is the value in Y that denotes the output or image of f corresponding to the input x.

The range (or image) of a function f:X → Y is the subset of Y, denoted f(X), defined by the following property:
y is an element of f(X) if and only if there is an x in X such that f(x) = y.
Use the definition to name functions. Do these examples as a whole class. Be sure to emphasize that students cannot define a function without also defining its domain and range. The definition talks about the output or image of f corresponding to the input x. The range is a subset of Y composed of the output or image values of f that correspond to each x in the domain X.

Exercises

Exercise 1.
Define f to assign each student at your school a unique ID number.
Assign each student a unique ID number.
a. Is this an example of a function? Use the definition to explain why or why not.
Yes, because each student in the school is assigned a unique student ID number. Every student only gets one ID number.

b. Suppose f(Hilda) = 350 123. What does that mean?
This means that Hilda’s ID number is 350,123.

c. Write your name and student ID number using function notation.
Solutions will vary but should follow the format f(Name) = Number.

Exercise 2.
Let g assign each student at your school to a grade level.
a. Is this an example of a function? Explain your reasoning.
Yes, this is a function because each student is assigned a single grade level. No students can be in both 9th and 10th grade.

b. Express this relationship using function notation, and state the domain and the range.
g: {students in the school} → {grade level}
Assign each student to a grade level.
Domain: All of the students enrolled in the school
Range: {9, 10, 11, 12}

Exercise 3.
Let h be the function that assigns each student ID number to a grade level.
Assign each student ID number to the student’s current grade level.
a. Describe the domain and range of this function.
Domain: the set of all numbers used as student IDs at my school
Range: {9, 10, 11, 12}

b. Record several ordered pairs (x, f(x)) that represent yourself and students in your group or class.
Solutions will vary but should be of the form (student ID number, grade level).

c. Jonny says, “This is not a function because every ninth grader is assigned the same range value of 9. The range only has 4 numbers {9, 10, 11, 12}, but the domain has a number for every student in our school.” Explain to Jonny why he is incorrect.
The definition of a function says each element in the domain is assigned to one element in the range. Assigning the same range value repeatedly does not violate the definition of a function. In fact, the situation would still be a function if there were only one element in the range.

### Eureka Math Algebra 1 Module 3 Lesson 9 Problem Set Answer Key

Question 1.
Which of the following are examples of a function? Justify your answers.
a. The assignment of the members of a football team to jersey numbers.
Yes. Each team member gets only one jersey number.

b. The assignment of U.S. citizens to Social Security numbers.
Yes. Each U.S. citizen who has applied for and received a Social Security number gets only one number.
Note: The domain is not necessarily all U.S. citizens, but those who applied for and received a SSN.

c. The assignment of students to locker numbers.
Yes. (The answer could be no if a student claims that certain students get assigned two or more lockers such as one for books and one for PE clothes.)

d. The assignment of the residents of a house to the street addresses.
Yes. People do not have more than one street address for the house in which they live.
Even if a person has more than one house, he or she only has one residence.

e. The assignment of zip codes to residences.
No. One zip code is assigned to multiple residences.

f. The assignment of residences to zip codes.
Yes. Each residence is assigned only one zip code.

g. The assignment of teachers to students enrolled in each of their classes.
No. Each teacher is assigned multiple students in each class.

h. The assignment of all real numbers to the next integer equal to or greater than the number.
Yes. Each real number is assigned to exactly one integer.

i. The assignment of each rational number to the product of its numerator and denominator.
No. While the product of any two numbers is a single number, there is no single way to write a rational number: $$\frac{1}{3}$$ = $$\frac{2}{6}$$ = $$\frac{3}{9}$$ = …, so there is no single product. A rational number in reduced form, i.e., the GCF of the numerator and denominator is 1, could be used to define a function.

Question 2.
Sequences are functions. The domain is the set of all term numbers (which is usually the positive integers), and the range is the set of terms of the sequence. For example, the sequence 1, 4, 9, 16, 25, 36, … of perfect squares is the function:
Let f:{positive integers}→{perfect squares}
Assign each term number to the square of that number.
a. What is f(3)? What does it mean?
f(3) = 9. It is the value of the 3rd square number. 9 dots can be arranged in a 3 by 3 square array.

b. What is the solution to the equation f(x) = 49? What is the meaning of this solution?
The solution is x = 7. It means that the 7th square number is 49, or the number 49 is the 7th term in the square number sequence.

c. According to this definition, is -3 in the domain of f? Explain why or why not.
No. The domain is the set of positive integers, and -3 is a negative number.

d. According to this definition, is 50 in the range of f? Explain why or why not.
It is not in the range of the function f because 50 is not a perfect square.

Question 3.
Write each sequence as a function.
Student responses to the following problems can vary. A sample solution is provided.

a. {1, 3, 6, 10, 15, 21, 28}
Let f:{1, 2, 3, 4, 5, 6, 7} → {1, 3, 6, 10, 15, 21, 28}
Assign each term number to the sum of the counting numbers from one to the term number.

b. {1, 3, 5, 7, 9,…}
Let f:{positive integers} → {positive odd integers}
Assign each positive integer to the number one less than double the integer.

c. a(n + 1) = 3an, a1 = 1, where n is a positive integer greater than or equal to 1.
Let f:{positive integers} → {positive integers}
Assign to each positive integer the value of 3 raised to the power of that integer minus 1.

### Eureka Math Algebra 1 Module 3 Lesson 9 Exit Ticket Answer Key

Question 1.
Given f as described below.
f:{whole numbers} → {whole numbers}
Assign each whole number to its largest place value digit.
For example, f(4) = 4, f(14) = 4, and f(194) = 9.
a. What is the domain and range of f?
Domain: all whole numbers; Range: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

b. What is f(257)?
7

c. What is f(0)?
0

d. What is f(999)?