Domain and Range 6.7

Definitions

• Domain: Defined as the set of all input values in a function. It is represented by using curly brackets {} to encapsulate the set of values.

• Range: Defined as the set of all output values in a function. Like the domain, it is also expressed using curly brackets {}.

Proper Notation

• It is essential to use the correct variables to denote the domain and range for clear communication of mathematical functions.

Examples of Domain and Range

Example 1: Area as a Function of Side Length

• Context: The area of a shape is dependent on its dimensions.

• Variable Definitions:

  • Let $s$ represent the side length.

  • Let $A$ represent the area.

• Domain:

  • $ext{Domain} = ext{{s > 0}}$

  • This implies that the side length must be greater than zero.

• Range:

  • $ext{Range} = ext{{A > 0}}$

  • This indicates that the area must also be greater than zero.

Example 2: Revenue as a Function of Number of Campers

• Context: Revenue is determined by the count of participants in a camping scenario.

• Variable Definitions:

  • Let $n$ be the number of campers.

  • Let $R$ be the amount of revenue generated.

• Domain:

  • $ext{Domain} = ext{{5n ≤ 16}}$

  • Where $n$ is constrained to be a whole number (non-negative integers).

• Range:

  • $ext{Range} = ext{{200 R ≤ 640}}$

  • This indicates that revenue must also be a multiple of 40.

Learning Targets

1. Understanding Function Domains and Ranges:

   • A student should comprehend the concept of domain and range as it pertains to any given function.

2. Determining Domain and Range from Situational Descriptions:

   • Given a scenario description, the student should be able to deduce reasonable input and output values for the function represented.

Warm-up Activity

Scenario:

• The total number of barks a dog has emitted is dependent on the time, measured in seconds, since its owner left the dog tied to a post.

• The owner returns less than 3 minutes later, hence students must evaluate the possible inputs (time) and outputs (number of barks) of such a situation.

Questions to Consider

1. Can the following values be valid inputs for the function?:

   • 84 seconds, 15 seconds, 300 seconds -> Yes for inputs of 84 and 15.

   • 300 seconds is invalid since it exceeds the time limit of 3 minutes (180 seconds).

2. Can the following values be valid outputs of the function?:

   • 15 barks, 84 barks, 300 barks.

     • 15 barks -> Yes.

     • 84 barks -> Condition-dependent. Cannot exceed the dog’s capacity of barking in the given timeframe.

     • 300 barks -> No. Not feasible within the limited time.

Possible or Impossible Inputs

Function: Area of a Square

• Given the function defined as $A(s) = s^2$ where $s$ represents the side length of the square (in cm).

Classification of Inputs:

Outputs of Functions

Function A: Area of a Square

1. Graph Representation: The equation $A(s) = s^2$ depicts an upward-sloping curve in a two-dimensional space where two axes denote the area in square centimeters versus the side length in centimeters.

2. Input-Output Pairs:

   • Selected pairs represented as coordinates:

     • (1,1)

     • (2,4)

     • (3,9)

     • (4,16)

3. Possible Output Values: All output values of function A are non-negative, thus defined as all numbers greater than or equal to zero.

Function R: Revenue for a Tennis Camp

1. Definition: $R(n) = 40n$ where $n$ represents the number of campers.

2. Evaluation of Values:

   • Is 20 a possible output?

     • $20 = 40(0.5)$, which cannot represent half a camper.

   • Is 100 a possible output?

     • $100 = 40(2.5)$, hence again invalid as it represents a fraction of a camper.

3. Graph Representation: Discussions around two graphs depict student counts against revenue. Students identify possible outputs and conclude that negative or fractional campers are not viable.

4. Output Values:

   • Set of all possible output values:

   • $ext{Range} = ext{{200 ≤ R ≤ 640}}$

   • specifies that R must be a multiple of 40.