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.

Students may ignore the function name and attend to only the input value. For instance, they may say, “ means that 60 seconds have passed.” Explain that the input value of 60, or , does represent that 60 seconds have passed, but the expression 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 of how to express the notation orally. Explain that the expression is read " of 60," is read " of 160," and is read " of ."

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 is read “ of .” It tells us that is the name of the function, is the input of the function, and is the output, or the value, of the function when the input is .
• The statement is read: “ of is equal to .” It tells us that is the name of the function and is the input. It also tells us that is the output, or the value, of the function at  and has the same value as .

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 a function, but isn’t?” (Each input for has a unique output, while inputs for 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 the rule  using function notation, for instance,  ? Why or why not?” (No. This notation is reserved for functions, and the input August 26 has many outputs, including mathematician Katherine Johnson, chemist Antoine Lavoisier, and many others.)

Some students might wonder if 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 , we would likely get different birthdays for the output.

Acknowledge that this is true, and that 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).