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

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Understanding the Concept of Inverse Functions

Inverse functions are a fundamental concept in mathematics, particularly in algebra and calculus. They play a crucial role in solving equations, graphing functions, and analyzing the behavior of functions. In this article, we will explore the concept of inverse functions and how to find the inverse of a given function.

What is an Inverse Function?

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. The inverse function is denoted by $f^{-1}(x)$.

Finding the Inverse of a Function

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

1. Replace $f(x)$ with $y$.
2. Swap the roles of $x$ and $y$.
3. Solve for $y$.

Finding the Inverse of $f(x)=\frac{1}{4} x-12$

Let's apply the steps above to find the inverse of $f(x)=\frac{1}{4} x-12$.

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

We start by replacing $f(x)$ with $y$, so we have:

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

Step 2: Swap the roles of $x$ and $y$

Next, we swap the roles of $x$ and $y$, so we have:

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

Step 3: Solve for $y$

Now, we need to solve for $y$. To do this, we can add 12 to both sides of the equation:

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

Next, we can multiply both sides of the equation by 4 to get rid of the fraction:

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

Expanding the left-hand side of the equation, we get:

$$4x+48 = y$$

So, the inverse function is:

$$h(x) = 4x+48$$

Conclusion

In this article, we explored the concept of inverse functions and how to find the inverse of a given function. We applied the steps above to find the inverse of $f(x)=\frac{1}{4} x-12$ and obtained the inverse function $h(x) = 4x+48$. This result is consistent with option D.

Final Answer

The final answer is:

• $h(x) = 4x+48$

This is the correct inverse function for $f(x)=\frac{1}{4} x-12$.

Q: What is the purpose of finding the inverse of a function?

A: The purpose of finding the inverse of a function is to reverse the operation of the original function. This is useful in solving equations, graphing functions, and analyzing the behavior of functions.

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, the function must pass the horizontal line test.

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. The function takes an input and produces an output, while its inverse takes the output and produces the original input.

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

A: To find the inverse of a function with a square root, you can follow these steps:

1. Replace the function with y.
2. Swap the roles of x and y.
3. Solve for y.

For example, if we have the function $f(x) = \sqrt{x-1}$, we can find its inverse by following these steps:

• Replace $f(x)$ with y: $y = \sqrt{x-1}$
• Swap the roles of x and y: $x = \sqrt{y-1}$
• Solve for y: $x^2 = y-1$, so $y = x^2 + 1$

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

A: To find the inverse of a function with a fraction, you can follow these steps:

1. Replace the function with y.
2. Swap the roles of x and y.
3. Solve for y.

For example, if we have the function $f(x) = \frac{1}{x+1}$, we can find its inverse by following these steps:

• Replace $f(x)$ with y: $y = \frac{1}{x+1}$
• Swap the roles of x and y: $x = \frac{1}{y+1}$
• Solve for y: $x(y+1) = 1$, so $xy + x = 1$, and $xy = 1 - x$, so $y = \frac{1-x}{x}$

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 it is denoted by $f^{-1}(x)$.

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

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

1. Graph the original function.
2. Reflect the graph of the original function across the line y = x.

For example, if we have the function $f(x) = x^2$, we can graph its inverse by reflecting the graph of $f(x) = x^2$ across the line y = x.

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. The function takes an input and produces an output, while its inverse takes the output and produces the original input.

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 function must pass the horizontal line test to have an inverse.

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

A: To check if a function is one-to-one, you can use the horizontal line test. If the function passes the horizontal line test, then it is one-to-one and has an inverse.

Q: What is the significance of the inverse of a function in real-world applications?

A: The inverse of a function is significant in real-world applications because it allows us to solve equations, graph functions, and analyze the behavior of functions. It is also used in many fields such as physics, engineering, and economics.

Q: Can a function have an inverse if it is not continuous?

A: No, a function cannot have an inverse if it is not continuous. The function must be continuous to have an inverse.

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

A: To find the inverse of a function with a logarithm, you can follow these steps:

1. Replace the function with y.
2. Swap the roles of x and y.
3. Solve for y.

For example, if we have the function $f(x) = \log(x+1)$, we can find its inverse by following these steps:

• Replace $f(x)$ with y: $y = \log(x+1)$
• Swap the roles of x and y: $x = \log(y+1)$
• Solve for y: $x = \log(y+1)$, so $e^x = y+1$, and $y = e^x - 1$

Q: Can a function have an inverse if it is not defined for all real numbers?

A: No, a function cannot have an inverse if it is not defined for all real numbers. The function must be defined for all real numbers to have an inverse.

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

A: To find the inverse of a function with a trigonometric function, you can follow these steps:

1. Replace the function with y.
2. Swap the roles of x and y.
3. Solve for y.

For example, if we have the function $f(x) = \sin(x)$, we can find its inverse by following these steps:

• Replace $f(x)$ with y: $y = \sin(x)$
• Swap the roles of x and y: $x = \sin(y)$
• Solve for y: $x = \sin(y)$, so $y = \sin^{-1}(x)$

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

A: No, a function cannot have an inverse if it is not one-to-one on a given interval. The function must be one-to-one on the given interval to have an inverse.

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

A: To find the inverse of a function with a rational function, you can follow these steps:

1. Replace the function with y.
2. Swap the roles of x and y.
3. Solve for y.

For example, if we have the function $f(x) = \frac{x+1}{x-1}$, we can find its inverse by following these steps:

• Replace $f(x)$ with y: $y = \frac{x+1}{x-1}$
• Swap the roles of x and y: $x = \frac{y+1}{y-1}$
• Solve for y: $x(y-1) = y+1$, so $xy - x = y + 1$, and $xy - y = x + 1$, so $y(x-1) = x + 1$, and $y = \frac{x+1}{x-1}$