Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

4.2.3 Finding a Unique Output for Each Input

4.2.3 • Finding a Unique Output for Each Input

Activity

Rule G takes a person's name as its input and gives their high school graduation date as the output.

Input	Output
Taylor Swift	2008
Zendaya	2015
Storm Reid	2020
Johnny Orlando	2021

Rule P takes a high school graduation date and gives a person with that graduation date as the output.

Input	Output
2020	Lil Tecca
2021	Olivia Rodrigo
2022	Forrest Wheeler
2023	Miles Brown

1. Complete the table for Rule G with three more examples of input-output pairs.

Compare your answers:

Your answer may vary, but here are some samples.

Input: Skai Jackson;

Output: 2020

Input: Keke Palmer;

Output: 2010

Your second input-output pair example:

Compare your answers:

Your answer may vary, but here are some samples.

Input: Jeremy Pena;

Output: 2015

Input: Mike Trout;

Output: 2009

Your third input-output pair example:

Compare your answers:

Your answer may vary, but here are some samples.

Input: Bryce Young;

Output: 2020

Input: Patrick Mahomes;

Output: 2014

2. Complete the table for Rule P with three more examples of input-output pairs

Compare your answer

Your answer may vary, but here are some samples.

Input: 2012;

Output: Ariana Grande

Input: 2012;

Output: Drake

Your second input-output pair example:

Compare your answers:

Your answer may vary, but here are some samples.

Input: 2020;

Output: Will Anderson, Jr.

Input: 2019;

Output: Kenyon Green

Input: 2015;

Output: Marisa Viggiano

Your third input-output pair example:

Compare your answers:

Your answer may vary, but here are some samples.

Input: 2018;

Output: Naomi Girma

Input: 2011;

Output: Lynn Williams

Input: 2018;

Output: Sophia Smith

3. If you use your own name as the input to G, explain how many outputs are possible and how you know.

Compare your answer:

Your answer may vary, but here is a sample.

One possible output. I can only graduate from high school once.

4. If you use your graduation date as the input to P, explain how many outputs are possible and how you know.

Compare your answer:

Your answer may vary, but here is a sample.

Many possible outputs. There are other people who will share the same graduation date as mine - especially my classmates who graduate with me.

5. Only one of these two relationships, G or P, is a function. The other is not a function. Which one is the function?

Compare your answers:

Rule G is a function and Rule P is not.

6. How do you know the relationship you chose is a function?

Compare your answers:

Your answer may vary, but here is a sample.

G only assigns one output to the input value.

7. For the relationship that is a function, write two input-output pairs from the table using function notation.

a. What is the first input-output pair example?

Compare your answers:

Your answer may vary, but here are some samples.

$G (T a y l o r S w i f t)$ = 2008

$G (S t o r m R e i d)$ = 2020

b. What is the second input-output pair example?

Compare your answers:

Your answer may vary, but here are some samples.

$G (Z e n d a y a)$ = 2015

$G (J o h n n y O r l a n d o)$ = 2021

Self Check

The price for pizzas, in dollars, is a function of the number of pizzas bought, $n$ .
Identify the input, output, and meaning of the statement: $p(11)=55$

Input: 11
Output: 55
55 pizzas cost $11
Input: 55
Output: 11
55 pizzas cost $11
Input: 11
Output: 55
11 pizzas cost $55
Input: 55
Output: 11
11 pizzas cost $55

Additional Resources

Meanings in Functions

Review the following examples to help you gain a better understanding of meanings in functions.

Example 1

The number of cubic yards of dirt, $D$ , needed to cover a garden with an area of $a$ square feet is given by $D = g (a)$ .

1. A garden with area 5000 $f t^{2}$ requires 50 $y d^{3}$ of dirt. Express this information in terms of the function $g$ .

Solution: $g (5000) = 50$

2. Explain the meaning of the statement $g (100) = 1$ .

Solution: The number of cubic yards of dirt required for a garden of 100 square feet is 1.

Example 2

Let $f (t)$ be the number of ducks in a lake $t$ years after 1990. Explain the meaning of each statement:

1. $f (5) = 30$

Solution: In 1995 (5 years after 1990), there were 30 ducks in the lake.

2. $f (10) = 40$

Solution: In 2000 (10 years after 1990), there were 40 ducks in the lake.

Try it

Meanings in Functions

The amount of garbage, $G$ , produced by a city with population $p$ is given by $G = f (p)$ . $G$ is measured in tons per week, and $p$ is measured in thousands of people.

1. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function $f$ .

Compare your answer:

$13 = f (40)$

2. Explain the meaning of the statement $f (5) = 2$ .

Compare your answer:

Five thousand people will produce 2 tons of garbage each week.

Are you ready for more?

Extending Your Thinking

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

$F$ (ONE) $= 3$

$F$ (TWO) $= 3$

$F$ (THREE) $= 5$

$F$ (FOUR) $= 4$

What rule could define function $F$ ?

Compare your answers:

Function $F$ takes a word for its input and gives the number of letters in that word for its output.

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

$v$ (ONE) $= 2$

$v$ (TWO) $= 1$

$v$ (THREE) $= 2$

$v$ (FOUR) $= 2$

What rule could define function $v$ ?

Compare your answers:

Function $v$ takes a word for its input and gives the number of vowels in that word as its output.