What Is The Inverse Of The Function $f(x) = \frac{1}{4}x - 12$?A. $h(x) = 48x - 4$B. $ H ( X ) = 48 X + 4 H(x) = 48x + 4 H ( X ) = 48 X + 4 [/tex]C. $h(x) = 4x - 48$D. $h(x) = 4x + 48$

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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 value of unknown variables.

Finding the Inverse of a Linear Function

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

1. Switch the variables: Swap the $x$ and $y$ variables in the original function.
2. Solve for $y$: Solve the resulting equation for $y$.
3. Interchange $x$ and $y$: Interchange the $x$ and $y$ variables to get the inverse function.

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: Switch the variables

Switch the $x$ and $y$ variables in the original function:

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

becomes

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

Step 2: Solve for $y$

Solve the resulting equation for $y$:

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

Add 12 to both sides:

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

Multiply both sides by 4:

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

Expand the left-hand side:

$$4x + 48 = y$$

Step 3: Interchange $x$ and $y$

Interchange the $x$ and $y$ variables to get the inverse function:

$$y = 4x + 48$$

becomes

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

Conclusion

The inverse of the function $f(x) = \frac{1}{4}x - 12$ is $h(x) = 4x + 48$. This means that if we apply the function $f(x)$ to a value $x$, and then apply the inverse function $h(x)$ to the result, we will get back the original value $x$.

Comparing the Options

Let's compare the options given in the problem:

A. $h(x) = 48x - 4$

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

C. $h(x) = 4x - 48$

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

Only option D matches the inverse function we found in the previous section.

Final Answer

The final answer is:

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

Understanding the Concept of Inverse Functions

In the previous article, we discussed the concept of inverse functions and how to find the inverse of a linear function. In this article, we will answer some frequently asked questions about inverse functions and provide examples to illustrate the concept.

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 and finding the value of unknown variables.

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

A: A function and its inverse are two different functions that "undo" each other. If we apply a function to a value and then apply its inverse to the result, we will get back the original value.

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

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

1. Switch the variables: Swap the $x$ and $y$ variables in the original function.
2. Solve for $y$: Solve the resulting equation for $y$.
3. Interchange $x$ and $y$: Interchange the $x$ and $y$ variables to get the inverse function.

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 a superscript $-1$.

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

A: Inverse functions are used in many real-world applications, such as:

• 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.
• Computer Science: Inverse functions are used in algorithms and data structures.

Example: Finding the Inverse of a Quadratic Function

Let's find the inverse of the quadratic function $f(x) = x^2 + 2x + 1$.

Step 1: Switch the variables

Switch the $x$ and $y$ variables in the original function:

$$y = x^2 + 2x + 1$$

becomes

$$x = y^2 + 2y + 1$$

Step 2: Solve for $y$

Solve the resulting equation for $y$:

$$x = y^2 + 2y + 1$$

Subtract 1 from both sides:

$$x - 1 = y^2 + 2y$$

Add 1 to both sides:

$$x - 1 + 1 = y^2 + 2y + 1$$

Simplify:

$$x = (y + 1)^2$$

Step 3: Interchange $x$ and $y$

Interchange the $x$ and $y$ variables to get the inverse function:

$$y = (x + 1)^2$$

becomes

$$h(x) = (x + 1)^2$$

Conclusion

In this article, we answered some frequently asked questions about inverse functions and provided examples to illustrate the concept. We also discussed the importance of inverse functions in real-world applications. By understanding the concept of inverse functions, we can solve equations and find the value of unknown variables.

Final Answer

The final answer is:

$$h(x) = (x + 1)^2$$