What Is The Inverse Of The Function F ( X ) = 1 4 X − 12 F(x)=\frac{1}{4} X-12 F ( X ) = 4 1 ​ X − 12 ?A. H ( X ) = 48 X − 4 H(x)=48 X-4 H ( X ) = 48 X − 4 B. H ( X ) = 48 X + 4 H(x)=48 X+4 H ( X ) = 48 X + 4 C. H ( X ) = 4 X − 48 H(x)=4 X-48 H ( X ) = 4 X − 48 D. H ( X ) = 4 X + 48 H(x)=4 X+48 H ( X ) = 4 X + 48

What is the Inverse of the Function $f(x)=\frac{1}{4} x-12$?

Understanding the Concept of Inverse Functions

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input. Inverse functions are denoted by a superscript $-1$ and are used to solve equations and find the values of unknown variables.

The Given Function

The given function is $f(x)=\frac{1}{4} x-12$. To find the inverse of this function, we need to follow a series of steps that will help us reverse the operation of the function.

Step 1: Replace $f(x)$ with $y$

The first step in finding the inverse of a function is to replace the function with a variable, usually $y$. This will help us simplify the function and make it easier to work with.

$$y = \frac{1}{4} x-12$$

Step 2: Swap $x$ and $y$

The next step is to swap the variables $x$ and $y$. This will help us reverse the operation of the function.

$$x = \frac{1}{4} y-12$$

Step 3: Solve for $y$

Now that we have swapped the variables, we need to solve for $y$. To do this, we will add $12$ to both sides of the equation and then multiply both sides by $4$.

$$x + 12 = \frac{1}{4} y$$

$$4(x + 12) = y$$

$$y = 4(x + 12)$$

Step 4: Simplify the Equation

Now that we have solved for $y$, we can simplify the equation by distributing the $4$ to the terms inside the parentheses.

$$y = 4x + 48$$

The Inverse Function

The inverse function of $f(x)=\frac{1}{4} x-12$ is $h(x)=4x+48$. This function takes the output of $f(x)$ and returns the original input.

Comparing the Options

Now that we have found the inverse function, we can compare it to the options given in the problem.

A. $h(x)=48 x-4$ B. $h(x)=48 x+4$ C. $h(x)=4 x-48$ D. $h(x)=4 x+48$

The only option that matches our answer is D. $h(x)=4 x+48$.

Conclusion

In this article, we have discussed the concept of inverse functions and how to find the inverse of a given function. We have used the function $f(x)=\frac{1}{4} x-12$ as an example and found its inverse function to be $h(x)=4x+48$. We have also compared our answer to the options given in the problem and found that option D is the correct answer.

What is the Inverse of the Function $f(x)=\frac{1}{4} x-12$?

The inverse of the function $f(x)=\frac{1}{4} x-12$ is $h(x)=4x+48$.

Why is the Inverse Function Important?

The inverse function is important because it allows us to solve equations and find the values of unknown variables. It is also used in many real-world applications, such as physics, engineering, and economics.

How to Find the Inverse of a Function

To find the inverse of a function, we need to follow a series of steps. The first step is to replace the function with a variable, usually $y$. The next step is to swap the variables $x$ and $y$. Then, we need to solve for $y$ and simplify the equation.

Common Mistakes When Finding the Inverse of a Function

One common mistake when finding the inverse of a function is to forget to swap the variables $x$ and $y$. Another mistake is to not simplify the equation after solving for $y$.

Real-World Applications of Inverse Functions

Inverse functions are used in many real-world applications, such as physics, engineering, and economics. For example, in physics, inverse functions are used to describe the motion of objects under the influence of gravity. In engineering, inverse functions are used to design and optimize systems. In economics, inverse functions are used to model the behavior of markets and economies.

Conclusion

In this article, we have discussed the concept of inverse functions and how to find the inverse of a given function. We have used the function $f(x)=\frac{1}{4} x-12$ as an example and found its inverse function to be $h(x)=4x+48$. We have also compared our answer to the options given in the problem and found that option D is the correct answer.
Q&A: Inverse Functions

Frequently Asked Questions About Inverse Functions

In this article, we will answer some of the most frequently asked questions about inverse functions. Whether you are a student, a teacher, or just someone who wants to learn more about inverse functions, this article is for you.

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 $f(x)$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input.

Q: Why is the inverse function important?

A: The inverse function is important because it allows us to solve equations and find the values of unknown variables. It is 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 a series of steps. The first step is to replace the function with a variable, usually $y$. The next step is to swap the variables $x$ and $y$. Then, you need to solve for $y$ and simplify the equation.

Q: What are some common mistakes when finding the inverse of a function?

A: One common mistake when finding the inverse of a function is to forget to swap the variables $x$ and $y$. Another mistake is to not simplify the equation after solving for $y$.

Q: Can you give an example of how to find the inverse of a function?

A: Let's say we have the function $f(x) = 2x + 3$. To find the inverse of this function, we need to follow the steps I mentioned earlier.

First, we replace the function with a variable, usually $y$.

$$y = 2x + 3$$

Next, we swap the variables $x$ and $y$.

$$x = 2y + 3$$

Then, we solve for $y$.

$$x - 3 = 2y$$

$$\frac{x - 3}{2} = y$$

Finally, we simplify the equation.

$$y = \frac{x - 3}{2}$$

So, the inverse of the function $f(x) = 2x + 3$ is $f^{-1}(x) = \frac{x - 3}{2}$.

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 output value corresponds to exactly one input value. In other words, if a function passes the horizontal line test, it has an inverse.

Q: Can you give an example of a function that does not have an inverse?

A: Let's say we have the function $f(x) = x^2$. This function does not have an inverse because it is not one-to-one. For example, the output value $f(2) = 4$ corresponds to two input values, $x = 2$ and $x = -2$.

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

A: Inverse functions are used in many real-world applications, such as physics, engineering, and economics. For example, in physics, inverse functions are used to describe the motion of objects under the influence of gravity. In engineering, inverse functions are used to design and optimize systems. In economics, inverse functions are used to model the behavior of markets and economies.

Q: How do I use inverse functions in real-world applications?

A: To use inverse functions in real-world applications, you need to understand how to find the inverse of a function and how to apply it to a specific problem. For example, if you are designing a system that needs to optimize its performance, you can use inverse functions to model the behavior of the system and find the optimal solution.

Conclusion

In this article, we have answered some of the most frequently asked questions about inverse functions. We have discussed what an inverse function is, why it is important, and how to find the inverse of a function. We have also given examples of how to find the inverse of a function and how to apply it to real-world applications. Whether you are a student, a teacher, or just someone who wants to learn more about inverse functions, this article is for you.