Which Function Is The Inverse Of $f(x) = 2x + 3$?A. $f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2}$B. $f^{-1}(x) = \frac{1}{2}x - \frac{3}{2}$C. $f^{-1}(x) = -2x + 3$D. $f^{-1}(x) = 2x + 3$

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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 original input x. In this article, we will explore the concept of inverse functions and identify the correct inverse of the given function f(x) = 2x + 3.

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 original input x. The inverse function is denoted by f^(-1)(x) and is read as "f inverse of x".

Properties of Inverse Functions

Inverse functions have several important properties that make them useful in mathematics. Some of the key properties of inverse functions include:

  • One-to-One Correspondence: An inverse function is a one-to-one correspondence between the input and output values of the original function.
  • Reversibility: An inverse function reverses the operation of the original function.
  • Symmetry: An inverse function is symmetric with respect to the line y = x.

Finding the Inverse of a Function

To find the inverse of a function, we need to follow a series of steps. Here are the steps to find the inverse of a function:

  1. Replace f(x) with y: The first step is to replace f(x) with y. This will help us to work with the function in terms of y instead of x.
  2. Interchange x and y: The next step is to interchange x and y. This will help us to get the inverse function in terms of x.
  3. Solve for y: The final step is to solve for y. This will give us the inverse function in terms of x.

Finding the Inverse of f(x) = 2x + 3

Now that we have understood the concept of inverse functions and the steps to find the inverse of a function, let's find the inverse of f(x) = 2x + 3.

Step 1: Replace f(x) with y

The first step is to replace f(x) with y. So, we get:

y = 2x + 3

Step 2: Interchange x and y

The next step is to interchange x and y. So, we get:

x = 2y + 3

Step 3: Solve for y

The final step is to solve for y. To do this, we need to isolate y on one side of the equation. We can do this by subtracting 3 from both sides of the equation and then dividing both sides by 2.

x - 3 = 2y (x - 3)/2 = y

So, the inverse function of f(x) = 2x + 3 is f^(-1)(x) = (x - 3)/2.

Conclusion

In this article, we have explored the concept of inverse functions and identified the correct inverse of the given function f(x) = 2x + 3. We have also discussed the properties of inverse functions and the steps to find the inverse of a function. We have found that the inverse function of f(x) = 2x + 3 is f^(-1)(x) = (x - 3)/2.

Answer

The correct answer is:

A. f^(-1)(x) = -\frac{1}{2}x - \frac{3}{2}

However, the correct answer is actually:

B. f^(-1)(x) = \frac{1}{2}x - \frac{3}{2}

This is because the inverse function of f(x) = 2x + 3 is f^(-1)(x) = (x - 3)/2, which is equivalent to f^(-1)(x) = \frac{1}{2}x - \frac{3}{2}.

Final Answer

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 on inverse functions, covering topics such as the definition of inverse functions, properties of inverse functions, and how to find the inverse of a function.

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 original input x.

Q: What are the properties of inverse functions?

A: Inverse functions have several important properties, including:

  • One-to-One Correspondence: An inverse function is a one-to-one correspondence between the input and output values of the original function.
  • Reversibility: An inverse function reverses the operation of the original function.
  • Symmetry: An inverse function is symmetric with respect to the line y = x.

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

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

  1. Replace f(x) with y: The first step is to replace f(x) with y. This will help you to work with the function in terms of y instead of x.
  2. Interchange x and y: The next step is to interchange x and y. This will help you to get the inverse function in terms of x.
  3. Solve for y: The final step is to solve for y. This will give you the inverse function in terms of x.

Q: What is the inverse of f(x) = 2x + 3?

A: To find the inverse of f(x) = 2x + 3, we need to follow the steps outlined above.

Step 1: Replace f(x) with y

y = 2x + 3

Step 2: Interchange x and y

x = 2y + 3

Step 3: Solve for y

x - 3 = 2y (x - 3)/2 = y

So, the inverse function of f(x) = 2x + 3 is f^(-1)(x) = (x - 3)/2.

Q: What is the inverse of f(x) = x^2?

A: To find the inverse of f(x) = x^2, we need to follow the steps outlined above.

Step 1: Replace f(x) with y

y = x^2

Step 2: Interchange x and y

x = y^2

Step 3: Solve for y

x = y^2 y = ±√x

So, the inverse function of f(x) = x^2 is f^(-1)(x) = ±√x.

Q: What is the inverse of f(x) = 1/x?

A: To find the inverse of f(x) = 1/x, we need to follow the steps outlined above.

Step 1: Replace f(x) with y

y = 1/x

Step 2: Interchange x and y

x = 1/y

Step 3: Solve for y

x = 1/y y = 1/x

So, the inverse function of f(x) = 1/x is f^(-1)(x) = 1/x.

Conclusion

In this article, we have provided a comprehensive Q&A guide on inverse functions, covering topics such as the definition of inverse functions, properties of inverse functions, and how to find the inverse of a function. We have also provided examples of how to find the inverse of various functions, including f(x) = 2x + 3, f(x) = x^2, and f(x) = 1/x.

Final Answer

The final answer is that inverse functions are a fundamental concept in mathematics, and understanding them is crucial for solving problems in various fields. By following the steps outlined in this article, you can find the inverse of any function and apply it to solve real-world problems.