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Chapter 2: Problem 7

Determine whether each function is one-to-one. $$\\{(-1,1),(1,1),(-2,4),(2,4)\\}$$

Short Answer

Expert verified
The function is not one-to-one.

Step by step solution

01

- Understanding One-to-One Function

A function is one-to-one if each element of the domain is mapped to a distinct element of the range. In other words, no two different domain elements map to the same range element.
02

- Analyzing the Given Function

Examine the pairs in the set \(\{(-1,1),(1,1),(-2,4),(2,4)\}\). Observe whether any two different domain elements map to the same range element.
03

- Identifying Repetitions

Notice that both \( -1 \) and \( 1 \) map to \( 1 \), and both \( -2 \) and \( 2 \) map to \( 4 \). This indicates that the same range elements are mapped from different domain elements.
04

- Conclusion

Since there are domain elements that map to the same range elements, the function \(\{(-1,1),(1,1),(-2,4),(2,4)\}\) is not one-to-one.

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Key Concepts

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

Domain and Range
In function analysis, the domain is the set of all possible input values (x-values) that a function can accept, while the range is the set of all possible output values (y-values) that the function can produce. For example, in the given function \(({-1,1),(1,1),(-2,4),(2,4)} \), the domain is \({ -1, 1, -2, 2 } \) because these are the x-values presented. The range is \({ 1, 4 } \) because these are the y-values that result from the inputs. Identifying domain and range is crucial for understanding and analyzing a function's behavior. When you look at a function, always ask, *What values can I put in?* (domain), and *What values can come out?* (range).
Function Analysis
Analyzing a function involves looking at how each element in the domain maps to an element in the range. This process helps to understand whether a function is one-to-one, onto, or neither. For the function at hand, \(({-1,1),(1,1),(-2,4),(2,4)} \), the first step is to list pairs of domain and range values and observe their relationship. You can see that \ (-1) \ maps to \ 1 \ and \ (1) \ also maps to the same value \ 1 \. Similarly, \ (-2) \ and \ (2) \ both map to \ 4 \. This repeated mapping of different domain elements to the same range element is a key indication in determining if a function meets specific criteria like being one-to-one.
Distinct Mappings
Distinct mappings refer to each input (domain value) mapping to a unique output (range value). A function is called one-to-one (or injective) if it meets this criterion. Failure to meet this criterion is revealed when different domain elements map to the same range element. For the given pairs \(({-1,1),(1,1),(-2,4),(2,4)} \), the fact that \ { -1, 1} \ both map to \ 1 \, and \ { -2, 2} \ both map to \ 4 \, shows that the function does not have distinct mappings, and is thus not one-to-one. This means we cannot match every x-value to a unique y-value, clearly indicating that the function fails the one-to-one test by redundancy in its mapped output values.

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