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Describing Functional Relationships

4.1.4 Describing Functional Relationships

4.1.4 • Describing Functional Relationships

Activity

In a function, there are two quantities called variables. The independent variable is the input, while the dependent variable is the output.

Here are two pairs of quantities from a situation you’ve seen in this lesson. Each pair has a relationship that can be defined as a function.

  • Time, in seconds, since the dog owner left and the total number of times the dog has barked
  • Time, in seconds, since the owner left and the total distance, in feet, that the dog has walked while waiting

Choose one pair of quantities and express their relationship as a function.

1. In that function, which variable is independent? Which one is dependent?

Compare your answers:

The time, in seconds, since the owner left is independent. The total number of times the dog has barked is dependent.

The time, in seconds, since the owner left is independent. The total distance, in feet, the dog walked while waiting is dependent.

2. Write a sentence of the form “____ is a function of ____.”

Compare your answers:

The total number of times the dog has barked is a function of the time, in seconds, since the owner left.

The total distance, in feet, is a function of the time, in seconds, since the owner left.

3. On a piece of graph paper, sketch a possible graph that represents your chosen scenario.

Compare your answer: Your answer may vary, but here is a sample for each of the scenarios. Time, in seconds, since the dog owner left and the total number of times the dog has barked

A scatter plot showing the total number of barks (y-axis) increasing over time in seconds (x-axis) from 0 to about 160 seconds, indicating a positive correlation between time and barks.

Time, in seconds, since the owner left and the total distance, in feet, that the dog has walked while waiting

Line graph showing total distance walked (in feet) increasing over time. The x-axis ranges from 0 to 160, and the y-axis ranges from 0 to 30. The line rises unevenly, with steeper increases at certain intervals.

Video: Determining Independent and Dependent Variables

Watch the following video to learn more about independent and dependent variables.

Access multimedia content

Self Check

The number of unread emails in Bryan’s account is 100. This number grows by 15 unread emails a day. Which of the following is true about this relationship?
  1. The number of unread emails is a function of the number of new unread emails.
  2. There is not a functional relationship between time and the number of emails.
  3. The number of emails is a function of time in days.
  4. Time, in days, is a function of the number of emails.

Additional Resources

Identifying if a Relationship Is a Function

A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

How to determine if a relationship is a function.

  1. Identify the input values.
  2. Identify the output values.
  3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

Example 1

The coffee shop menu, shown below, consists of items and their prices.

A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.

a. Is price a function of the item?

Compare your answer:

Let’s begin by considering the input values as the items on the menu. The output values are then the prices. Each item on the menu has only one price, so the price is a function of the item.

b. Is the item a function of the price?

Compare your answer:

Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See the image below. Therefore, the item is a not a function of price.

Association of the prices to the donuts.

Example 2

In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? The table below shows a possible rule for assigning grade points.

Percent grade 0–56 57–61 62–66 67–71 72–77 78–86 87–91 92 – 100
Grade point average 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0
This table outlines a grading scale, showing how specific percentage grade ranges correspond to their respective grade point averages.

Solution

For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.

In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades, ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

Try it

Identifying if a Relationship Is a Function

The table below lists the five greatest baseball players of all time in order of rank.

Player Rank
Babe Ruth 1
Willie Mays 2
Ty Cobb 3
Walter Johnson 4
Hank Aaron 5

a. Is the rank a function of the player name?

Compare your answers:

Here is how to determine which variable is a function of the other:

Identify the input and output.

Since the name is the input and the rank is the output, the rank is a function of the player name.

b. Is the player name a function of the rank?

Compare your answers:

Here is how to determine which variable is a function of the other:

Identify the input and output.

Since the rank is the input and the name is the output, the player name is a function of the rank.