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Chapter 3: Problem 34

(a) use roster notation to represent the domain. (b) use roster notation to represent the range. \(\\{\) (orange, 72 calories), (peach, 37 calories), (apple, 72 calories), (banana, 105 calories)\\}

Short Answer

Expert verified
Domain: \(\{ \text{orange}, \text{peach}, \text{apple}, \text{banana} \} \); Range: \(\{ 72, 37, 105 \} \)

Step by step solution

01

Identify the Domain

The domain is the set of all the first components in the ordered pairs. Look at each ordered pair and list all the unique items (or fruits in this case).
02

Represent the Domain in Roster Notation

Write down the domain using roster notation. The domain is the list of fruits: \(\{ \text{orange}, \text{peach}, \text{apple}, \text{banana} \} \)
03

Identify the Range

The range is the set of all the second components in the ordered pairs. Look at each ordered pair and list all the unique calorie values.
04

Represent the Range in Roster Notation

Write down the range using roster notation. The range is the list of calorie values: \(\{ 72, 37, 105 \} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain
The domain of a function or a relation is an essential concept in mathematics. It refers to the set of all possible input values. In our example, the input values are the fruits.
Here's how we identified the domain step-by-step:

  • We have the ordered pairs: (orange, 72 calories), (peach, 37 calories), (apple, 72 calories), and (banana, 105 calories).
  • The domain includes all the first components of these ordered pairs, which means every unique fruit in our list.

So, the domain using roster notation is \( \{ \text{orange}, \text{peach}, \text{apple}, \text{banana} \} \). Roster notation means listing the elements of a set between braces \{ \}, separated by commas.
Range
The range is another key concept in functions and relations. It refers to the set of all possible output values. In our example, the output values are the calories.
To find the range, follow these steps:

  • Again, we look at the ordered pairs: (orange, 72 calories), (peach, 37 calories), (apple, 72 calories), and (banana, 105 calories).
  • The range includes all the second components of these ordered pairs, which means every unique calorie count in our list.

Thus, the range in roster notation is \( \{72, 37, 105\} \). Note that 72 is listed only once even though it appears twice. Roster notation helps us express these values clearly and concisely.
Ordered Pairs
Understanding ordered pairs is fundamental to working with domains and ranges. An ordered pair is a pair of elements used to show a relationship between them.
In our example, the ordered pairs are (fruit, calories):

  • (orange, 72 calories)
  • (peach, 37 calories)
  • (apple, 72 calories)
  • (banana, 105 calories)

Each ordered pair consists of two parts:

  • The first component, here the fruit, represents the input value.
  • The second component, in this case, the calorie count, represents the output value.

Ordered pairs are useful in various mathematical applications, such as plotting points on a graph or establishing clear relationships in data sets.

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