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Chapter 6: Problem 17

Determine whether the function is one-to-one. $$ \\{(2,6),(-3,6),(4,9),(1,10)\\} $$

Short Answer

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

Step by step solution

01

Understand the concept of one-to-one functions

A function is one-to-one if every element of the range is mapped from a unique element of the domain. This means no two different elements in the domain map to the same element in the range.
02

Identify the domain and range

List the domain and range from the given set of ordered pairs. Domain: \(\{2, -3, 4, 1\}\) Range: \(\{6, 9, 10\}\)
03

Check for repeated range values

Look at the range values and see if any value is repeated. In this case, '6' appears twice in the range: (2,6) and (-3,6).
04

Determine if the function is one-to-one

Since the value '6' in the range is mapped from two different elements in the domain (2 and -3), the function 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.

Function Mapping
Function mapping is the process through which each element in the domain of a function is paired with exactly one element in the range. Think of it as assigning unique partners to everyone at a dance party.
In a function, the domain consists of all the possible inputs (or x-values), and the range consists of all the possible outputs (or y-values).
For instance, in the given set \{(2,6),(-3,6),(4,9),(1,10)\},
the domain is \( \{2, -3, 4, 1\} \) and the range is \( \{6, 9, 10\} \).
Domain
The domain of a function includes all the possible inputs or x-values that can be mapped to outputs in the range. It's like a guest list at an exclusive party; only those included can participate.
In the context of our set \{(2,6),(-3,6),(4,9),(1,10)\},
the domain is \( \{2, -3, 4, 1\} \). These x-values are the points from which the function derives its outputs.
Whenever you try to determine the domain, simply list all unique x-values from your ordered pairs.
Range
The range involves all possible outputs (or y-values) that a function can map to. In simpler terms, it's the results you get from applying the function to each element in the domain.
For the set \{(2,6),(-3,6),(4,9),(1,10)\}, the range is \( \{6, 9, 10\} \).
Notice that even though the function is applied to multiple x-values, we only list each y-value once to form the range. It's important to check for unique values here, especially when determining if a function is one-to-one.
Unique Elements
Unique elements are crucial in determining if a function is one-to-one. A function is considered one-to-one if each element in the range is paired with only one unique element in the domain.
In simpler terms, no two different x-values should map to the same y-value.
In our example, since the y-value '6' is paired with both '2' and '-3', the function is not one-to-one.
Checking for these unique mappings helps solidify your understanding that one-to-one functions require each output to be distinctively linked to a single input.

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Most popular questions from this chapter

In Problems 71-78, use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{3} 21\)

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