Find The Inverse Of The Following Function:$\[ F(x) = \frac{1}{2} X - 5 \\]A. $\[ F^{-1}(x) = 2x + 10 \\]B. $\[ F^{-1}(x) = \frac{1}{2} X + \frac{5}{2} \\]C. $\[ F^{-1}(x) = 2x + 5 \\]D. $\[ F^{-1}(x) = 2x -

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Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. The inverse of a function essentially reverses the operation of the original function, providing a new function that undoes the original function's action. In this article, we will explore how to find the inverse of a linear function, specifically the function f(x)=12x−5f(x) = \frac{1}{2}x - 5. We will examine the process of finding the inverse and evaluate the given options to determine the correct answer.

What is a Linear Function?

A linear function is a type of function that can be written in the form f(x)=mx+bf(x) = mx + b, where mm is the slope of the line and bb is the y-intercept. The graph of a linear function is a straight line, and the function can be represented by a linear equation. In the given function f(x)=12x−5f(x) = \frac{1}{2}x - 5, the slope is 12\frac{1}{2} and the y-intercept is −5-5.

Finding the Inverse of a Linear Function

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

  1. Switch the x and y variables: The first step in finding the inverse of a function is to switch the x and y variables. This means that we will replace xx with yy and yy with xx in the original function.
  2. Solve for y: Once we have switched the x and y variables, we need to solve for y. This involves isolating y on one side of the equation.
  3. Interchange x and y: Finally, we interchange x and y to obtain the inverse function.

Finding the Inverse of f(x)=12x−5f(x) = \frac{1}{2}x - 5

Let's apply the steps outlined above to find the inverse of the given function f(x)=12x−5f(x) = \frac{1}{2}x - 5.

Step 1: Switch the x and y variables

Switching the x and y variables, we get:

x=12y−5x = \frac{1}{2}y - 5

Step 2: Solve for y

To solve for y, we need to isolate y on one side of the equation. We can do this by adding 5 to both sides of the equation and then multiplying both sides by 2.

x+5=12yx + 5 = \frac{1}{2}y

2(x+5)=y2(x + 5) = y

2x+10=y2x + 10 = y

Step 3: Interchange x and y

Finally, we interchange x and y to obtain the inverse function:

f−1(x)=2x+10f^{-1}(x) = 2x + 10

Evaluating the Options

Now that we have found the inverse of the given function, let's evaluate the options provided:

A. f−1(x)=2x+10f^{-1}(x) = 2x + 10

B. f−1(x)=12x+52f^{-1}(x) = \frac{1}{2}x + \frac{5}{2}

C. f−1(x)=2x+5f^{-1}(x) = 2x + 5

D. f−1(x)=2x−5f^{-1}(x) = 2x - 5

Based on our calculations, we can see that option A is the correct answer.

Conclusion

In this article, we explored how to find the inverse of a linear function, specifically the function f(x)=12x−5f(x) = \frac{1}{2}x - 5. We followed the steps outlined above to find the inverse and evaluated the given options to determine the correct answer. The inverse of the given function is f−1(x)=2x+10f^{-1}(x) = 2x + 10. This demonstrates the importance of understanding inverse functions in mathematics and how they can be used to solve problems.

Additional Examples

To further illustrate the concept of finding the inverse of a linear function, let's consider a few more examples:

  • Example 1: Find the inverse of the function f(x)=3x−2f(x) = 3x - 2.
  • Example 2: Find the inverse of the function f(x)=2x+1f(x) = 2x + 1.
  • Example 3: Find the inverse of the function f(x)=x−4f(x) = x - 4.

By following the steps outlined above and applying them to these examples, we can see that the process of finding the inverse of a linear function is a straightforward one.

Final Thoughts

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about inverse functions. Whether you are a student looking for help with homework or a teacher seeking to clarify concepts for your students, this Q&A article is designed to provide you with the information you need.

Q: What is an inverse function?

A: An inverse function is a function that undoes the action of the original function. 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: Why are inverse functions important?

A: Inverse functions are important because they allow us to solve equations and find the value of a function at a specific point. They are also used in many real-world applications, such as physics, engineering, and economics.

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. Switch the x and y variables.
  2. Solve for y.
  3. Interchange x and y.

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

A: A function and its inverse are two different functions that are related to each other. The function f(x) and its inverse f^(-1)(x) are two different functions that are inverses of each other.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse of a function is unique and is denoted by f^(-1)(x).

Q: How do I know if a function has an inverse?

A: A function has an inverse if it is one-to-one, meaning that each value of x corresponds to a unique value of y. If a function is not one-to-one, it does not have an inverse.

Q: What is the notation for the inverse of a function?

A: The notation for the inverse of a function is f^(-1)(x).

Q: Can I use a calculator to find the inverse of a function?

A: Yes, you can use a calculator to find the inverse of a function. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

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

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

  1. Switch the x and y variables.
  2. Graph the resulting function.

Q: Can I use the inverse of a function to solve equations?

A: Yes, you can use the inverse of a function to solve equations. If you have an equation of the form f(x) = y, you can use the inverse of f(x) to solve for x.

Q: What are some real-world applications of inverse functions?

A: Inverse functions have many real-world applications, including:

  • Physics: Inverse functions are used to describe the motion of objects under the influence of forces.
  • Engineering: Inverse functions are used to design and optimize systems.
  • Economics: Inverse functions are used to model the behavior of economic systems.

Conclusion

In this article, we have addressed some of the most frequently asked questions about inverse functions. We have covered topics such as the definition of an inverse function, how to find the inverse of a function, and the notation for the inverse of a function. We have also discussed some real-world applications of inverse functions. Whether you are a student or a teacher, we hope that this article has provided you with the information you need to understand inverse functions.