Which Of The Following Is The Inverse Of $y = 121$?A. $y = -\log 1, X$B. $yhinl$C. \$y = \log, 12$[/tex\]D. $4/Imatix$

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Introduction

In mathematics, an inverse function is a function that reverses the operation of another function. It is denoted by the symbol ^{-1} and is used to find the value of the input that produces a given output. In this article, we will explore the concept of inverse functions and identify the correct inverse of the given function y = 121.

What is an Inverse Function?

An inverse function is a function that undoes the operation of another function. It is a one-to-one function that takes the output of the original function and returns the input that produced that output. In other words, if f(x) is a function and f^{-1}(x) is its inverse, then f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

Types of Inverse Functions

There are two types of inverse functions: one-to-one and onto functions. A one-to-one function is a function that maps each input to a unique output, while an onto function is a function that maps every possible output to at least one input.

Properties of Inverse Functions

Inverse functions have several properties that make them useful in mathematics. Some of these properties include:

  • Symmetry: If f(x) is a function and f^{-1}(x) is its inverse, then f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.
  • One-to-one: Inverse functions are one-to-one functions, meaning that each input maps to a unique output.
  • Onto: Inverse functions are onto functions, meaning that every possible output maps to at least one input.

Finding the Inverse of a Function

To find the inverse of a function, we need to follow these steps:

  1. Replace f(x) with y: Replace the function f(x) with y to make it easier to work with.
  2. Interchange x and y: Interchange the x and y variables to get x = f(y).
  3. Solve for y: Solve the equation x = f(y) for y to get the inverse function.

Example: Finding the Inverse of y = 121

To find the inverse of the function y = 121, we need to follow the steps above.

  1. Replace f(x) with y: Replace the function f(x) with y to get y = 121.
  2. Interchange x and y: Interchange the x and y variables to get x = 121.
  3. Solve for y: Solve the equation x = 121 for y to get y = 121.

However, this is not the correct inverse of the function y = 121. The correct inverse is a function that takes the output of the original function and returns the input that produced that output.

Analyzing the Options

Let's analyze the options given to identify the correct inverse of the function y = 121.

A. y = -log 1, x: This option is not a valid inverse function. The function y = -log 1, x is not a one-to-one function, and it does not satisfy the property of symmetry.

B. yhinl: This option is not a valid inverse function. The function yhinl is not a one-to-one function, and it does not satisfy the property of symmetry.

C. y = log, 12: This option is a valid inverse function. The function y = log, 12 is a one-to-one function, and it satisfies the property of symmetry.

D. 4/Imatix: This option is not a valid inverse function. The function 4/Imatix is not a one-to-one function, and it does not satisfy the property of symmetry.

Conclusion

In conclusion, the correct inverse of the function y = 121 is y = log, 12. This function satisfies the properties of an inverse function, including symmetry, one-to-one, and onto.

References

Introduction

Inverse functions are a fundamental concept in mathematics, and understanding them is crucial for solving problems in various fields, including algebra, calculus, and engineering. In this article, we will provide a comprehensive Q&A guide to inverse functions, covering topics such as the definition, properties, and examples of inverse functions.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. It is denoted by the symbol ^{-1} and is used to find the value of the input that produces a given output.

Q: What are the properties of an inverse function?

A: The properties of an inverse function include:

  • Symmetry: If f(x) is a function and f^{-1}(x) is its inverse, then f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.
  • One-to-one: Inverse functions are one-to-one functions, meaning that each input maps to a unique output.
  • Onto: Inverse functions are onto functions, meaning that every possible output maps to at least one input.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, follow these steps:

  1. Replace f(x) with y: Replace the function f(x) with y to make it easier to work with.
  2. Interchange x and y: Interchange the x and y variables to get x = f(y).
  3. Solve for y: Solve the equation x = f(y) for y to get the inverse function.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two different functions that work together to produce the same output. The function takes the input and produces the output, while the inverse function takes the output and produces the input.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse function is unique and is denoted by the symbol ^{-1}.

Q: How do I determine if a function is one-to-one?

A: To determine if a function is one-to-one, check if each input maps to a unique output. If each input maps to a unique output, then the function is one-to-one.

Q: What is the importance of inverse functions in real-life applications?

A: Inverse functions are used in various real-life applications, including:

  • Calculus: Inverse functions are used to find the derivative and integral of a function.
  • Engineering: Inverse functions are used to design and analyze systems, such as electrical circuits and mechanical systems.
  • Computer Science: Inverse functions are used in algorithms and data structures, such as sorting and searching.

Q: Can you provide an example of finding the inverse of a function?

A: Yes, let's find the inverse of the function y = 2x + 3.

  1. Replace f(x) with y: Replace the function f(x) with y to get y = 2x + 3.
  2. Interchange x and y: Interchange the x and y variables to get x = 2y + 3.
  3. Solve for y: Solve the equation x = 2y + 3 for y to get y = (x - 3) / 2.

The inverse function is y = (x - 3) / 2.

Conclusion

In conclusion, inverse functions are a fundamental concept in mathematics, and understanding them is crucial for solving problems in various fields. This Q&A guide provides a comprehensive overview of inverse functions, including their definition, properties, and examples. We hope this guide has been helpful in understanding inverse functions and their applications.

References