Select The Correct Answer.Which Function Is The Inverse Of Function $f$?$f(x)=\frac{x+2}{7}$A. $s(x)=2x+7$B. $q(x)=\frac{-x+2}{7}$C. $r(x)=\frac{7}{x+2}$D. $p(x)=7x-2$

Introduction

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.

What is an Inverse Function?

An inverse function is a function that undoes the action of another function. For example, if we have a function $f(x) = 2x + 3$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input. In this case, the inverse function is $f^{-1}(x) = \frac{x-3}{2}$.

How to Find 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 $x$ and $y$.
3. Solve for $y$.

Let's apply these steps to the given function $f(x) = \frac{x+2}{7}$.

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

$f(x) = \frac{x+2}{7}$

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

$y = \frac{x+2}{7}$

Step 2: Swap $x$ and $y$

Swap $x$ and $y$:

$x = \frac{y+2}{7}$

Step 3: Solve for $y$

Solve for $y$:

$x = \frac{y+2}{7}$

Multiply both sides by 7:

$7x = y + 2$

Subtract 2 from both sides:

$7x - 2 = y$

Now, we have the inverse function:

$f^{-1}(x) = 7x - 2$

Comparing the Options

Now that we have found the inverse function, let's compare it with the given options:

A. $s(x) = 2x + 7$

B. $q(x) = \frac{-x+2}{7}$

C. $r(x) = \frac{7}{x+2}$

D. $p(x) = 7x - 2$

The correct answer is:

D. $p(x) = 7x - 2$

This is because the inverse function we found is $f^{-1}(x) = 7x - 2$, which matches option D.

Conclusion

$$$$

Introduction

Inverse functions are a fundamental concept in mathematics, and understanding them is crucial for solving equations and finding the values of unknown variables. In this article, we will answer some frequently asked questions about inverse functions, providing a comprehensive guide to help you understand this concept better.

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: How do I find the inverse of a function?

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

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

Let's illustrate this with an example. Suppose we have a function $f(x) = 2x + 3$. To find its inverse, we would:

1. Replace $f(x)$ with $y$: $y = 2x + 3$
2. Swap $x$ and $y$: $x = 2y + 3$
3. Solve for $y$: $x - 3 = 2y$, so $y = \frac{x-3}{2}$

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. For example, if we have a function $f(x) = 2x + 3$, its inverse function $f^{-1}(x) = \frac{x-3}{2}$ will take the output of $f(x)$ and return the original input.

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 a function have more than one inverse?

A: No, a function can only have one inverse. The inverse of a function is unique and is denoted by a superscript $-1$.

Q: How do I use inverse functions to solve equations?

A: To use inverse functions to solve equations, you need to follow these steps:

1. Write the equation in function notation.
2. Find the inverse of the function.
3. Use the inverse function to solve for the unknown variable.

Let's illustrate this with an example. Suppose we have the equation $y = 2x + 3$ and we want to solve for $x$. We can use the inverse function $f^{-1}(x) = \frac{x-3}{2}$ to solve for $x$:

$y = 2x + 3$

$f^{-1}(y) = \frac{y-3}{2}$

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

Now, we can solve for $x$:

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

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

$x = \frac{2x}{2}$

$x = x$

Therefore, the solution to the equation is $x = \frac{y-3}{2}$.

Q: What are some common applications of inverse functions?

A: Inverse functions have many applications in mathematics, science, and engineering. Some common applications include:

• Solving equations and finding the values of unknown variables.
• Modeling real-world phenomena, such as population growth and decay.
• Analyzing data and making predictions.
• Solving optimization problems.

Conclusion

In this article, we answered some frequently asked questions about inverse functions, providing a comprehensive guide to help you understand this concept better. We discussed how to find the inverse of a function, the difference between a function and its inverse, and how to use inverse functions to solve equations. We also explored some common applications of inverse functions.