Find The Inverse Of F ( X ) = − 4 X − 12 F(x) = -4x - 12 F ( X ) = − 4 X − 12 .A. F − 1 ( X ) = 4 X − 3 F^{-1}(x) = 4x - 3 F − 1 ( X ) = 4 X − 3 B. F − 1 ( X ) = − 4 X − 3 F^{-1}(x) = -4x - 3 F − 1 ( X ) = − 4 X − 3 C. F − 1 ( X ) = − 1 4 X − 3 F^{-1}(x) = -\frac{1}{4}x - 3 F − 1 ( X ) = − 4 1 ​ X − 3 D. F − 1 ( X ) = 1 4 X + 3 F^{-1}(x) = \frac{1}{4}x + 3 F − 1 ( X ) = 4 1 ​ X + 3

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Introduction

In mathematics, the concept of an inverse function is crucial in understanding the relationship between two functions. The inverse of a function essentially reverses the operation of the original function. In this article, we will focus on finding the inverse of a linear function, specifically the function f(x)=−4x−12f(x) = -4x - 12. We will explore the steps involved in finding the inverse and provide a clear explanation of the process.

What is a Linear Function?

A linear function is a polynomial function of degree one, which means it has the form f(x)=ax+bf(x) = ax + b, where aa and bb are constants. The graph of a linear function is a straight line. In the case of the function f(x)=−4x−12f(x) = -4x - 12, the coefficient of xx is −4-4, and the constant term is −12-12.

The Inverse of a Linear Function

To find the inverse of a linear function, we need to follow a series of steps. The first step is to replace f(x)f(x) with yy to simplify the notation. So, we have y=−4x−12y = -4x - 12. The next step is to swap the roles of xx and yy, which gives us x=−4y−12x = -4y - 12. Now, we need to solve for yy in terms of xx.

Solving for y

To solve for yy, we need to isolate yy on one side of the equation. We can start by adding 1212 to both sides of the equation, which gives us x+12=−4yx + 12 = -4y. Now, we can divide both sides of the equation by −4-4 to solve for yy. This gives us y=−14x−3y = -\frac{1}{4}x - 3.

Conclusion

In conclusion, the inverse of the linear function f(x)=−4x−12f(x) = -4x - 12 is f−1(x)=−14x−3f^{-1}(x) = -\frac{1}{4}x - 3. This means that if we apply the function ff to a value xx, and then apply the inverse function f−1f^{-1} to the result, we will get back the original value xx.

Answer

The correct answer is:

C. f−1(x)=−14x−3f^{-1}(x) = -\frac{1}{4}x - 3

Example

Let's consider an example to illustrate the concept of an inverse function. Suppose we have the function f(x)=2x+1f(x) = 2x + 1. To find the inverse of this function, we can follow the same steps as before. We start by replacing f(x)f(x) with yy to simplify the notation. So, we have y=2x+1y = 2x + 1. Next, we swap the roles of xx and yy, which gives us x=2y+1x = 2y + 1. Now, we need to solve for yy in terms of xx.

To solve for yy, we can start by subtracting 11 from both sides of the equation, which gives us x−1=2yx - 1 = 2y. Now, we can divide both sides of the equation by 22 to solve for yy. This gives us y=12x−12y = \frac{1}{2}x - \frac{1}{2}.

Final Answer

In conclusion, the inverse of the linear function f(x)=2x+1f(x) = 2x + 1 is f−1(x)=12x−12f^{-1}(x) = \frac{1}{2}x - \frac{1}{2}.

Why is the Inverse of a Linear Function Important?

The inverse of a linear function is important because it helps us to understand the relationship between two functions. In many real-world applications, we need to find the inverse of a function to solve a problem or to model a situation. For example, in physics, the inverse of a function can be used to model the motion of an object. In economics, the inverse of a function can be used to model the relationship between two variables.

How to Find the Inverse of a Linear Function

To find the inverse of a linear function, we need to follow a series of steps. The first step is to replace f(x)f(x) with yy to simplify the notation. So, we have y=ax+by = ax + b. The next step is to swap the roles of xx and yy, which gives us x=ay+bx = ay + b. Now, we need to solve for yy in terms of xx.

To solve for yy, we can start by subtracting bb from both sides of the equation, which gives us x−b=ayx - b = ay. Now, we can divide both sides of the equation by aa to solve for yy. This gives us y=1ax−bay = \frac{1}{a}x - \frac{b}{a}.

Conclusion

In conclusion, finding the inverse of a linear function is an important concept in mathematics. It helps us to understand the relationship between two functions and is used in many real-world applications. By following the steps outlined in this article, we can find the inverse of a linear function and use it to solve problems or model situations.

Final Answer

The final answer is:

C. f−1(x)=−14x−3f^{-1}(x) = -\frac{1}{4}x - 3

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Frequently Asked Questions

Q: What is the inverse of a linear function?

A: The inverse of a linear function is a function that reverses the operation of the original function. In other words, if we apply the original function to a value x, and then apply the inverse function to the result, we will get back the original value x.

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

A: To find the inverse of a linear function, we need to follow a series of steps. The first step is to replace f(x) with y to simplify the notation. So, we have y = ax + b. The next step is to swap the roles of x and y, which gives us x = ay + b. Now, we need to solve for y in terms of x.

Q: What is the formula for the inverse of a linear function?

A: The formula for the inverse of a linear function is y = (1/a)x - (b/a), where a and b are the coefficients of the original function.

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

A: Yes, you can use a calculator to find the inverse of a linear function. Most graphing calculators have a built-in function to find the inverse of a function.

Q: What is the difference between the inverse of a linear function and the original function?

A: The inverse of a linear function is a function that reverses the operation of the original function. In other words, if we apply the original function to a value x, and then apply the inverse function to the result, we will get back the original value x. The original function and its inverse are symmetric with respect to the line y = x.

Q: Can I find the inverse of a non-linear function?

A: No, you cannot find the inverse of a non-linear function in the same way that you can find the inverse of a linear function. Non-linear functions do not have an inverse that is also a function.

Q: What is the importance of finding the inverse of a linear function?

A: Finding the inverse of a linear function is important because it helps us to understand the relationship between two functions. In many real-world applications, we need to find the inverse of a function to solve a problem or to model a situation.

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

A: Yes, you can use the inverse of a linear function to solve a system of equations. If you have a system of equations with two variables, you can use the inverse of one of the functions to solve for one of the variables.

Q: What is the relationship between the inverse of a linear function and the original function?

A: The inverse of a linear function is a function that reverses the operation of the original function. In other words, if we apply the original function to a value x, and then apply the inverse function to the result, we will get back the original value x. The original function and its inverse are symmetric with respect to the line y = x.

Q: Can I graph the inverse of a linear function?

A: Yes, you can graph the inverse of a linear function. The graph of the inverse of a linear function is a reflection of the graph of the original function across the line y = x.

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

A: The graph of a linear function is a straight line, while the graph of its inverse is a reflection of the graph of the original function across the line y = x.

Q: Can I use the inverse of a linear function to model a real-world situation?

A: Yes, you can use the inverse of a linear function to model a real-world situation. For example, you can use the inverse of a linear function to model the relationship between the cost of a product and the number of units sold.

Q: What is the importance of understanding the inverse of a linear function?

A: Understanding the inverse of a linear function is important because it helps us to understand the relationship between two functions. In many real-world applications, we need to find the inverse of a function to solve a problem or to model a situation.

Q: Can I use the inverse of a linear function to solve a problem in physics?

A: Yes, you can use the inverse of a linear function to solve a problem in physics. For example, you can use the inverse of a linear function to model the motion of an object.

Q: What is the relationship between the inverse of a linear function and the original function in terms of the slope?

A: The inverse of a linear function has a slope that is the reciprocal of the slope of the original function.

Q: Can I use the inverse of a linear function to solve a problem in economics?

A: Yes, you can use the inverse of a linear function to solve a problem in economics. For example, you can use the inverse of a linear function to model the relationship between the price of a product and the quantity demanded.

Q: What is the importance of understanding the inverse of a linear function in terms of the y-intercept?

A: Understanding the inverse of a linear function in terms of the y-intercept is important because it helps us to understand the relationship between the original function and its inverse. The y-intercept of the inverse function is the reciprocal of the y-intercept of the original function.