# Lesson 2

Function Notation

## 2.1: Back to the Post! (10 minutes)

### Warm-up

The goal of this warm-up is to motivate the need for a notation that can be used to communicate about functions.

Students analyze three graphs from an earlier lesson, interpret various points on the graphs, and use their analyses to answer questions about the situations. This work requires students to make careful connections between points on the graphs, pairs of input and output values, and verbal descriptions of the functions. Students find that, unless each feature and the function being referenced is clearly articulated, which could be tedious to do, what they wish to communicate about the functions may be ambiguous or unclear.

When answering the last two questions, students are likely to find the prompts lacking in specificity and to probe: “for which day?” Suggest that they answer based on their interpretation of the questions.

Then, look for students who assume that the questions refer to one particular function and those who assume they refer to all three functions (and consequently answer them for each function). Ask them to share their interpretations during the whole-class discussion.

### Student Facing

Here are the graphs of some situations you saw before. Each graph represents the distance of a dog from a post as a function of time since the dog owner left to purchase something from a store. Distance is measured in feet and time is measured in seconds.

Day 1

Day 2

Day 3

1. Use the given graphs to answer these questions about each of the three days:

1. How far away was the dog from the post 60 seconds after the owner left?

Day 1:

Day 2:

Day 3:

2. How far away was the dog from the post when the owner left?

Day 1:

Day 2:

Day 3:

3. The owner returned 160 seconds after he left. How far away was the dog from the post at that time?

Day 1:

Day 2:

Day 3:

4. How many seconds passed before the dog reached the farthest point it could reach from the post?

Day 1:

Day 2:

Day 3:

2. Consider the statement, “The dog was 2 feet away from the post after 80 seconds.” Do you agree with the statement?
3. What was the distance of the dog from the post 100 seconds after the owner left?

### Activity Synthesis

Invite students to share their response to the first set of questions.

To help illustrate that it could be tedious to refer to a specific part of a function fully and precisely, ask each question completely for each of the three days. (For instance, “How far away was the dog from the post 60 seconds after the owner left on Day 1? How far away was the dog from the post 60 seconds after the owner left on Day 2?”) If students offer a numerical value (for instance, “1.5 feet”) without stating what question it answers or to what quantity it corresponds to, ask them to clarify.

Next, select previously identified students to share their responses to the last two questions. Regardless of whether students chose to answer them for a particular day or for all three days, point out that the answers depend on the day. When the day (or the function) is not specified, it is unclear what information is sought.

Explain that sometimes we need to be pretty specific when talking about functions. But to be specific could require many words and become burdensome. Tell students that they will learn about a way to describe functions clearly and succinctly.

## 2.2: A Handy Notation (15 minutes)

### Activity

In this activity, students learn that function notation can be used as a handy shorthand for communicating about functions and specific parts or features of a function. They interpret statements that are written in this notation and use the notation to refer to points on a graph or to represent simple verbal statements about a function.

### Launch

Explain to students that one way to talk about functions precisely and without wordy descriptions is by naming the functions and using function notation.

• Suppose we give a name to each function that relates the dog’s distance from the post and the time since the dog owner left: function $$f$$ for Day 1, function $$g$$ for Day 2, function $$h$$ for Day 3. The input of each function is time in seconds, $$t$$
• To represent “the distance of the dog from the post 60 seconds after the owner left,” we can simply write: $$f(60)$$. To express the same quantity for the second and third day, we can write $$g(60)$$ and $$h(60)$$

Ask students to refer to the three graphs from the warm-up to answer the questions.

### Student Facing

Let’s name the functions that relate the dog’s distance from the post and the time since its owner left: function $$f$$ for Day 1, function $$g$$ for Day 2, function $$h$$ for Day 3. The input of each function is time in seconds, $$t$$.

1. Use function notation to complete the table.
day 1 day 2 day 3
a. distance from post 60 seconds after the owner left
b. distance from post when the owner left
c. distance from post 150 seconds after the owner left
2. Describe what each expression represents in this context:

1. $$f(15)$$
2. $$g(48)$$
3. $$h(t)$$
3. The equation $$g(120) = 4$$ can be interpreted to mean: “On Day 2, 120 seconds after the dog owner left, the dog was 4 feet from the post.”

What does each equation mean in this situation?

1. $$h(40) = 4.6$$
2. $$f(t) = 5$$
3. $$g(t) = d$$

### Anticipated Misconceptions

Students may ignore the function name and attend only to the input value. For instance, they may say “$$f(60)$$ means that 60 seconds have passed.” Explain that the input value of 60 or $$t=60$$ does represents that 60 seconds have passed, but the expression $$f(60)$$ represents the output value of the function. In this case, it means the dog’s distance from the post, on Day 1, 60 seconds after its owner left.

### Activity Synthesis

Invite students to share their responses. As students begin to share, they may be unsure as to how to express the notation orally. Explain that the expression $$f(60)$$ is read "$$f$$ of 60," $$g(150)$$ is read "$$g$$ of 150," and $$h(t)$$ is read "$$h$$ of $$t$$."

To make sure students see the structure of this new notation, consider displaying it and annotating each part, as shown here.

Clarify that:

• The notation $$f(x)$$ is read “$$f$$ of $$x$$.” It tells us that $$f$$ is the name of the function, $$x$$ is the input of the function, and $$f(x)$$ is the output or the value of the function when the input is $$x$$.
• The statement $$g(t) = d$$ is read: “$$g$$ of $$t$$ is equal to $$d$$.” It tells us that $$g$$ is the name of the function and $$t$$ is the input. It also tells us that $$g(t)$$ is the output or the value of the function at $$t$$, and $$g(t)$$ has the same value as $$d$$.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their description of what each expression represents in context, present an incorrect response and explanation. For example, “$$f(15)$$ is the input value. It tells us that 15 seconds have passed.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, monitor for students who clarify that $$f(15)$$ represents an output value, specifically the distance of the dog from the post 15 seconds after the owner left on day 1. This helps students evaluate, and improve upon, the written mathematical arguments of others, as they interpret function notation in context.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of function notation. Provide students with a physical copy of the annotated function notation.
Supports accessibility for: Conceptual processing; Language

## 2.3: Birthdays (10 minutes)

### Activity

This activity reinforces students’ understanding about what makes a relationship between two variables a function, namely, that it gives a unique output for each input. It also prompts students to use function notation to express a functional relationship that does not involve numerical values for its input and output.

### Student Facing

Rule $$B$$ takes a person’s name as its input, and gives their birthday as the output.

input output
Abraham Lincoln February 12

Rule $$P$$ takes a date as its input and gives a person with that birthday as the output.

input output
August 26 Katherine Johnson

1. Complete each table with three more examples of input-output pairs.
2. If you use your name as the input to $$B$$, how many outputs are possible? Explain how you know.
3. If you use your birthday as the input to $$P$$, how many outputs are possible? Explain how you know.
4. Only one of the two relationships is a function. The other is not a function. Which one is which? Explain how you know.
5. For the relationship that is a function, write two input-output pairs from the table using function notation.

### Student Facing

#### Are you ready for more?

1. Write a rule that describes these input-output pairs:

$$F(\text{ONE})=3$$

$$F(\text{TWO})=3$$

$$F(\text{THREE})=5$$

$$F(\text{FOUR})=4$$

2. Here are some input-output pairs with the same inputs but different outputs:

$$v(\text{ONE})=2$$

$$v(\text{TWO})=1$$

$$v(\text{THREE})=2$$

$$v(\text{FOUR})=2$$

What rule could define function $$v$$?

### Anticipated Misconceptions

If a student wonders what happens to a person born on February 29, tell them that the output of the function is the original birth date, not the annual birthday.

### Activity Synthesis

Discuss with students:

• “Why is $$B$$ a function, but $$P$$ isn’t?” (Each input for $$B$$ has a unique output, while inputs for $$P$$ may have several outputs. For example, March 14 is the birthday of Albert Einstein, Stephen Curry, Billy Crystal, Simone Biles and many other people. February 12 is the birthday of Abraham Lincoln and Charles Darwin.)
• “Would it be acceptable to express relationship $$P$$ using function notation, for instance,: $$P(\text{August 26})= \text{Katherine Johnson}$$? Why or why not?” (No, because this notation is reserved for functions.)

Some students might wonder if $$B$$ is still a function if multiple people have the same name. For instance, there might be a few people named Katherine Johnson, and if we enter “Katherine Johnson” as the input for $$B$$, we would likely get different birthdays for the output.

Acknowledge that this is true, and that $$B$$ would only be a function if it assumes that no two people have the same full name, or if another identifier could be used to tell apart people with the same first name and last name (for instance, if a middle name or initial is also used, or if a number is added to each Katherine Johnson to distinguish them from one another).

Conversing: MLR2 Collect and Display. Listen for and collect vocabulary, gestures, and diagrams students use to describe what makes a relationship between two variables a function. Capture student language that reflects a variety of ways to describe the characteristics of and differences between a relationship that is a function and a relationship that is not a function. Amplify the words “unique,” “input,” and “output.” Write the students’ words on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

## Lesson Synthesis

### Lesson Synthesis

Refer back to the bagel shop activity from the opening lesson. Invite students to consider some statements about the function they saw in that situation. Display the following for all to see:

The best price for bagels, in dollars, is a function of the number of bagels bought, $$n$$.

$$b(2)$$

$$b(6)$$​​​​​​

$$b(11) = 10.50$$

$$b(13) = 11.25$$

Arrange students in groups of 2. Ask partners to take turns reading and interpreting the statements in function notation. Each person should:

• Read the statement aloud to their partner.
• Identify the input, the output, and the function in the statement.
• Explain the meaning of the entire statement using a complete sentence.

If students say that the first two statements have no outputs, clarify that both $$b(2)$$ and $$b(6)$$ represent outputs, even though the value of each is not stated.

## Student Lesson Summary

### Student Facing

Here are graphs of two functions, each representing the cost of riding in a taxi from two companies—Friendly Rides and Great Cabs.

For each taxi, the cost of a ride is a function of the distance traveled. The input is distance in miles, and the output is cost in dollars.

• The point $$(2,5.70)$$ on one graph tells us the cost of riding a Friendly Rides taxi for 2 miles.
• The point $$(2, 4.25)$$ on the other graph tells us the cost of riding a Great Cabs taxi for 2 miles.

We can convey the same information much more efficiently by naming each function and using function notation to specify the input and the output.

• Let’s name the function for Friendly Rides function $$f$$.
• Let's name the function for Great Cabs function $$g$$.
• To refer to the cost of riding each taxi for 2 miles, we can write: $$f(2)$$ and $$g(2)$$.
• To say that a 2-mile trip with Friendly Rides will cost \$5.70, we can write $$f(2)=5.70$$. • To say that a 2-mile trip with Great Cabs will cost \$4.25, we can write $$g(2)=4.25$$.

In general, function notation has this form:

It is read “$$f$$ of $$x$$” and can be interpreted to mean: $$f(x)$$ is the output of a function $$f$$ when $$x$$ is the input.