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Determining if menu price lists are functions

The coffee shop menu, shown in [link] consists of items and their prices.

  1. Is price a function of the item?
  2. Is the item a function of the price?
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.
  1. Let’s begin by considering the input 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.

  2. 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 [link] .
    Association of the prices to the donuts.

    Therefore, the item is a not a function of price.

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Determining if class grade rules are functions

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? [link] 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

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.

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[link] http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014. 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

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

a. yes; b. yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

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Using function notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into graphing calculators and computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables h for height and a for age. The letters f , g , and h are often used to represent functions just as we use x , y , and z to represent numbers and A , B , and C to represent sets.

h  is  f  of  a We name the function  f ;  height is a function of age . h = f ( a ) We use parentheses to indicate the function input f ( a ) We name the function  f ;  the expression is read as “ f  of  a .”

Remember, we can use any letter to name the function; the notation h ( a ) shows us that h depends on a . The value a must be put into the function h to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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