Lesson 2

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.

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.

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.

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.

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.

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 \(d\).

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.

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.

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).