Problem 12 If the function is one-to-one, f... [FREE SOLUTION]

Chapter 12: Problem 12

If the function is one-to-one, find its inverse. \(\\{(-8,6),(-4,3),(0,6),(5,10)\\}\)

Short Answer

Expert verified

The function is not one-to-one.

Step by step solution

Verify Function is One-to-One

A function is one-to-one if each output is assigned to exactly one input. Check each output value in the function set to ensure no duplicates. Here, the outputs are \(6, 3, 6, 10\). Since \(6\) appears twice, this is not a one-to-one function.

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Key Concepts

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

One-to-One Function

A one-to-one function, also known as an injective function, has a specific property: each input maps to a unique output, and no output value is repeated.

If a function is one-to-one, it means that no two different inputs will produce the same output.
This property is crucial when determining the inverse function because it guarantees that each output has exactly one input.

To check if a function is one-to-one, you look at all output values and ensure there are no duplicates. In the given example, the function set is \( \{(-8,6),(-4,3),(0,6),(5,10)\} \). Checking the outputs \(6, 3, 6, 10\), we notice that \(6\) appears twice. Due to this repetition, the function is not one-to-one.
Since the function is not one-to-one, it does not have an inverse. Remember, for a function to have an inverse, it must be one-to-one.

Function Verification

Verifying whether a function is one-to-one is an essential step in determining if an inverse exists.

First, list all the output values.
Second, check for any repeated values.

In our set \( \{(-8,6),(-4,3),(0,6),(5,10)\} \), verifying involves making sure none of the output values are the same. However, we find that \(6\) is repeated.
This simple check tells us that the function fails the one-to-one criteria. If any output is repeated for different inputs, the function cannot have an inverse.

Output Values

Output values are the results you get after applying the function to each input. They play a key role in verifying the nature of the function.

For a function to be one-to-one, each output value must be unique.
If even one output value repeats for different inputs, the function is not one-to-one.

In the example \( \{(-8,6),(-4,3),(0,6),(5,10)\} \), the output values are \(6, 3, 6, 10\). Because \(6\) appears more than once, we know the function is not one-to-one.
Understanding and analyzing output values will help you determine whether a function has an inverse and how it behaves.
Always remember, unique output values are essential for a function to be one-to-one.

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91影视

If the function is one-to-one, find its inverse. \(\\{(-8,6),(-4,3),(0,6),(5,10)\\}\)

Short Answer

Step by step solution

Verify Function is One-to-One

Key Concepts

One-to-One Function

Function Verification

Output Values

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