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Chapter 4: Problem 53

If the function is one-to-one, find its inverse. $$\\{(1,-3),(2,-7),(4,-3),(5,-5)\\}$$

Short Answer

Expert verified
The function does not have an inverse because it is not one-to-one.

Step by step solution

01

Check for One-to-One Function

A function is one-to-one if each output is paired with exactly one input. In the given set, check if each output value is unique.
02

Identify Non-Unique Outputs

Examine the output values \(-3, -7, -3, -5\). Notice that -3 appears twice. Thus, this function is not one-to-one.
03

Conclusion

Since the function is not one-to-one, it does not have an inverse.

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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 function is called a one-to-one function if every element of the range is paired with exactly one element of the domain. This means that each output corresponds to only one input, making the relationship unique. For a visual example, imagine a set of lockers where every student has their own locker, and no two students share the same locker. If any locker is shared by multiple students, the function is not one-to-one.
This concept is crucial because only one-to-one functions can have inverses. If a function is not one-to-one, it cannot be reversed uniquely, making it impossible to find an inverse function.
Unique Outputs
Unique outputs in a function ensure that each output value is assigned solely one input value.
In the exercise provided, the function relates several pairs: \{(1,-3),(2,-7),(4,-3),(5,-5)\}. When we examine these pairs, we notice that the output \(-3\) appears twice, linked to inputs \(1\) and \(4\).
This repetition indicates that the function is not one-to-one, since -3 is the output for two different input values. Therefore, for a function to be one-to-one, each output value must be different and unique.
Function Properties
Understanding function properties is essential for determining if a function is one-to-one and for finding its inverse.
Key properties to look at include:
  • Domain: The set of all possible input values.
  • Range: The set of all possible output values.
  • Injective (One-to-One): Each output is linked to only one input.
  • Inverse: A function that reverses the original function.
If the function from the exercise had been one-to-one, each output value would correspond to a unique input value. Then, to find the inverse, you would simply swap the input-output pairs. However, because the function reflects a pair with the same output (i.e., -3 appearing twice), it lacks the injective property. Hence, it cannot have an inverse function.

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Most popular questions from this chapter

Emissions Tax One action that government could take to reduce carbon emissions into the atmosphere is to levy a tax on fossil fuel. This tax would be based on the amount of carbon dioxide emitted into the air when the fuel is burned. The cost-benefit equation $$\ln (1-P)=-0.0034-0.0053 T$$ models the approximate relationship between a tax of \(T\) dollars per ton of carbon and the corresponding percent reduction \(P\) (in decimal form) of emissions of carbon dioxide. (Source: Nordhause, W., "To Slow or Not to Slow: The Economics of the Greenhouse Effect," Yale University, New Haven, Connecticut.) (a) Write \(P\) as a function of \(T\). (b) Graph \(P\) for \(0 \leq T \leq 1000 .\) Discuss the benefit of continuing to raise taxes on carbon (c) Determine \(P\) when \(T=60\) dollars, and interpret this result. (d) What value of \(T\) will give a \(50 \%\) reduction in carbon emissions?

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