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Chapter 1: Problem 44

Suppose \(f\) is a function whose domain equals \\{2,4,7,8,9\\} and whose range equals \(\\{-3,0,2,6\\} .\) Explain why \(f\) is not a one-to-one function.

Short Answer

Expert verified
Function \(f\) with given domain \(\{2, 4, 7, 8, 9\}\) and range \(\{-3, 0, 2, 6\}\) can't be a one-to-one function because there are 5 elements in the domain and only 4 in the range. Since there are fewer elements in the range, at least two elements in the domain must share the same output value, which violates the one-to-one function criterion.

Step by step solution

01

Understand the given domain and range

Given, the domain of function \(f\) is \(\{2, 4, 7, 8, 9\}\), this means that there are 5 different input values for the function, and the range of function \(f\) is \(\{-3, 0, 2, 6\}\), which suggests that there are 4 distinct output values for the function.
02

Check if the function might be one-to-one

In a one-to-one function, every element in the domain must map to a unique element in the range. Since there are 5 elements in the domain and only 4 elements in the range, it's not possible for every element in the domain to map to a unique element in the range. There must be at least one element in the range that is mapped to by multiple elements of the domain.
03

Conclusion

Based on our analysis in step 2, it is clear that function \(f\) cannot be a one-to-one function since there are fewer elements in its range than in its domain. This means that at least two elements in the domain must share the same output value in the range, which violates the one-to-one function criterion. Thus, \(f\) is not a one-to-one function.

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