Math Problem Statement

Determine if the following function is one-to-one, onto, both, or neither based on the provided diagram.

Solution

To determine whether the given function is one-to-one, onto, both, or neither, let's analyze the mapping between set XX and set YY based on the diagram:

1. One-to-One (Injective):

A function is one-to-one if every element in the domain XX is mapped to a unique element in the codomain YY. In other words, no two different elements in XX should map to the same element in YY.

Looking at the diagram:

  • AFA \mapsto F
  • BGB \mapsto G
  • CHC \mapsto H
  • DHD \mapsto H
  • EIE \mapsto I

From this, we observe that both CC and DD map to the same element HH, which violates the one-to-one condition. Therefore, the function is not one-to-one.

2. Onto (Surjective):

A function is onto if every element in the codomain YY is mapped by at least one element from the domain XX. In other words, for every element in YY, there exists at least one element in XX that maps to it.

Looking at the elements in YY:

  • FF has a pre-image (mapped from AA).
  • GG has a pre-image (mapped from BB).
  • HH has pre-images (mapped from CC and DD).
  • II has a pre-image (mapped from EE).

Since every element in YY has at least one pre-image from XX, the function is onto.

Conclusion:

  • The function is not one-to-one (because CC and DD both map to HH).
  • The function is onto (because every element in YY is mapped by at least one element in XX).

Thus, the correct answer is: Onto.


Would you like more details or have any questions about this?

Here are 5 related questions to explore:

  1. What is the difference between injective and surjective functions?
  2. How can we test if a function is bijective?
  3. Can a function be surjective but not injective? Give an example.
  4. What are some real-world applications of one-to-one and onto functions?
  5. How do we represent a function algebraically compared to graphically?

Tip: In a diagram, if two elements from the domain map to the same element in the codomain, the function is not one-to-one (injective).

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Math Problem Analysis

Mathematical Concepts

Functions
Injective Function (One-to-One)
Surjective Function (Onto)

Formulas

-

Theorems

Definition of One-to-One (Injective) Function
Definition of Onto (Surjective) Function

Suitable Grade Level

Grades 9-12