Math Problem Statement

State the domain and range for the following relation. Then determine whether the relation represents a function.

Solution

Let's analyze the problem in the image.

Problem:

The problem asks to:

  1. Identify the correct domain and range for a given relation.
  2. Determine whether the relation represents a function.

First, the domain and range:

The provided diagram illustrates a relation between two sets:

  • Set 1 (Domain): It contains plant types such as "Shrub", "Daisy", and "Herb".
  • Set 2 (Range): It contains names of plants such as "Basil", "Rose", "Oak", and "Flower".

Mapping:

The arrows in the diagram indicate the relationship between elements of the domain and range:

  • Shrub maps to Basil.
  • Shrub also maps to Rose.
  • Daisy maps to Flower.
  • Herb maps to Oak.
  • Tree maps to Oak.

Analyzing the domain and range:

  • Domain: The elements that are being mapped from: Shrub, Daisy, Herb, Tree.
  • Range: The elements that are being mapped to: Basil, Rose, Flower, Oak.

So, the correct domain is: {Shrub,Daisy,Herb,Tree}\{\text{Shrub}, \text{Daisy}, \text{Herb}, \text{Tree}\}.
The correct range is: {Basil,Rose,Flower,Oak}\{\text{Basil}, \text{Rose}, \text{Flower}, \text{Oak}\}.

Determining whether this is a function:

A relation is a function if every element in the domain is mapped to exactly one element in the range.

  • Shrub is mapped to both Basil and Rose, which means one element in the domain is mapped to more than one element in the range.

Thus, this relation is not a function.

Conclusion:

The correct answer would be:

  • Domain: {Shrub,Daisy,Herb,Tree}\{\text{Shrub}, \text{Daisy}, \text{Herb}, \text{Tree}\}
  • Range: {Basil,Rose,Flower,Oak}\{\text{Basil}, \text{Rose}, \text{Flower}, \text{Oak}\}
  • The relation does not represent a function because an element in the first set (Shrub) corresponds to more than one element in the second set.

This corresponds to the following options in the image:

  • Domain: Shrub, Daisy, Herb, Tree.
  • Range: Basil, Rose, Flower, Oak.
  • Option B: "No, because each element in the first set corresponds to exactly one element in the second set."

Would you like more details on how relations and functions work?

Here are 5 related questions to expand on this:

  1. What are the differences between a relation and a function?
  2. How can you determine if a relation is a function from its graph?
  3. What are examples of real-world functions?
  4. Can a function have multiple outputs for a single input?
  5. How do you represent functions using set notation?

Tip: A function assigns exactly one output to each input. If any input has more than one output, the relation is not a function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Relations
Functions
Set Theory

Formulas

-

Theorems

Definition of Function
Set and Relations Theory

Suitable Grade Level

Grades 7-9