Complete The Inverse Function For The Function Given Below.If $f(x) = 2x - 4$, Then $f^{-1}(x) = \square X + \square$.Type The Correct Answer In Each Box. Use Numerals Instead Of Words. If Necessary, Use / For The Fraction Bar(s).

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Introduction

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x. In this article, we will focus on finding the inverse function for a given function.

What is an Inverse Function?

An inverse function is a function that undoes the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x. The inverse function is denoted by f^(-1)(x) and is read as "f inverse of x".

Why is Inverse Function Important?

Inverse functions are important in mathematics because they help us to solve equations and find the value of unknown variables. Inverse functions are also used in many real-world applications, such as physics, engineering, and economics.

How to Find the Inverse Function?

To find the inverse function of a given function f(x), we need to follow these steps:

  1. Switch x and y: Switch the x and y variables in the function f(x) to get the equation y = f(x).
  2. Interchange x and y: Interchange the x and y variables in the equation y = f(x) to get the equation x = f(y).
  3. Solve for y: Solve the equation x = f(y) for y to get the inverse function f^(-1)(x).

Example: Finding the Inverse Function of f(x) = 2x - 4

Let's find the inverse function of the function f(x) = 2x - 4.

Step 1: Switch x and y

Switch the x and y variables in the function f(x) = 2x - 4 to get the equation y = 2x - 4.

Step 2: Interchange x and y

Interchange the x and y variables in the equation y = 2x - 4 to get the equation x = 2y - 4.

Step 3: Solve for y

Solve the equation x = 2y - 4 for y to get the inverse function f^(-1)(x).

x = 2y - 4

Add 4 to both sides of the equation:

x + 4 = 2y

Divide both sides of the equation by 2:

(x + 4) / 2 = y

Therefore, the inverse function f^(-1)(x) is:

f^(-1)(x) = (x + 4) / 2

Conclusion

In this article, we have discussed the concept of inverse functions and how to find the inverse function of a given function. We have also provided an example of finding the inverse function of the function f(x) = 2x - 4. The inverse function f^(-1)(x) is a powerful tool in mathematics that helps us to solve equations and find the value of unknown variables.

Final Answer

The final answer is:

Q&A: Inverse Functions

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x.

Q: Why is inverse function important?

A: Inverse functions are important in mathematics because they help us to solve equations and find the value of unknown variables. Inverse functions are also used in many real-world applications, such as physics, engineering, and economics.

Q: How to find the inverse function?

A: To find the inverse function of a given function f(x), we need to follow these steps:

  1. Switch x and y: Switch the x and y variables in the function f(x) to get the equation y = f(x).
  2. Interchange x and y: Interchange the x and y variables in the equation y = f(x) to get the equation x = f(y).
  3. Solve for y: Solve the equation x = f(y) for y to get the inverse function f^(-1)(x).

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

A: A function and its inverse are two different functions that undo each other's operation. In other words, if we have a function f(x) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

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 f^(-1)(x).

Q: How to check if a function is one-to-one?

A: A function is one-to-one if it passes the horizontal line test. In other words, if we draw a horizontal line on the graph of the function, it should intersect the graph at most once.

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

A: The relationship between a function and its inverse is that they are two different functions that undo each other's operation. In other words, if we have a function f(x) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Q: Can a function have an inverse if it is not one-to-one?

A: No, a function cannot have an inverse if it is not one-to-one. The inverse function is unique and is denoted by f^(-1)(x).

Q: How to find the inverse of a composite function?

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

  1. Find the inverse of each function: Find the inverse of each function in the composite function.
  2. Compose the inverses: Compose the inverses of each function to get the inverse of the composite function.

Q: What is the difference between a function and its inverse in terms of domain and range?

A: The domain and range of a function and its inverse are swapped. In other words, if we have a function f(x) and its inverse f^(-1)(x), then the domain of f(x) is the range of f^(-1)(x) and the range of f(x) is the domain of f^(-1)(x).

Conclusion

In this article, we have discussed the concept of inverse functions and answered some common questions related to inverse functions. We have also provided examples and explanations to help you understand the concept of inverse functions.

Final Answer

The final answer is:

  • An inverse function is a function that reverses the operation of another function.
  • Inverse functions are important in mathematics because they help us to solve equations and find the value of unknown variables.
  • To find the inverse function of a given function f(x), we need to follow these steps: Switch x and y, interchange x and y, and solve for y.
  • A function and its inverse are two different functions that undo each other's operation.
  • A function cannot have more than one inverse.
  • A function is one-to-one if it passes the horizontal line test.
  • The relationship between a function and its inverse is that they are two different functions that undo each other's operation.
  • A function cannot have an inverse if it is not one-to-one.
  • To find the inverse of a composite function, we need to find the inverse of each function and compose the inverses.
  • The domain and range of a function and its inverse are swapped.