Math Problem Statement
Solution
To determine whether the given function is one-to-one, onto, both, or neither, let's analyze the mapping between set and set based on the diagram:
1. One-to-One (Injective):
A function is one-to-one if every element in the domain is mapped to a unique element in the codomain . In other words, no two different elements in should map to the same element in .
Looking at the diagram:
From this, we observe that both and map to the same element , 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 is mapped by at least one element from the domain . In other words, for every element in , there exists at least one element in that maps to it.
Looking at the elements in :
- has a pre-image (mapped from ).
- has a pre-image (mapped from ).
- has pre-images (mapped from and ).
- has a pre-image (mapped from ).
Since every element in has at least one pre-image from , the function is onto.
Conclusion:
- The function is not one-to-one (because and both map to ).
- The function is onto (because every element in is mapped by at least one element in ).
Thus, the correct answer is: Onto.
Would you like more details or have any questions about this?
Here are 5 related questions to explore:
- What is the difference between injective and surjective functions?
- How can we test if a function is bijective?
- Can a function be surjective but not injective? Give an example.
- What are some real-world applications of one-to-one and onto functions?
- 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