3. Which Of The Following Is The Inverse Of $f(x)=\frac{2x-3}{5}$?A. $f^{-1}(x)=\frac{5x+3}{2}$B. $f^{-1}(x)=\frac{-2x+3}{5}$C. $f^{-1}(x)=\frac{2y-3}{5}$D. $f^{-1}(x)=\frac{N^{\Omega}y-3}{2}$

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. The inverse of a function essentially reverses the operation of the original function, allowing us to find the input value that corresponds to a given output value. In this article, we will explore how to find the inverse of a function, using the given function $f(x)=\frac{2x-3}{5}$ as an example.

What is an Inverse Function?

An inverse function is a function that undoes the operation of the original 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 value. The inverse function is denoted by $f^{-1}(x)$.

Step 1: Replace f(x) with y

To find the inverse of a function, we start by replacing the function $f(x)$ with a variable $y$. This will allow us to work with the function in a more manageable way.

Step 2: Swap x and y

Next, we swap the variables $x$ and $y$. This is a crucial step in finding the inverse function, as it allows us to work with the function in a way that is opposite to the original function.

Step 3: Solve for y

Now, we need to solve for $y$ in terms of $x$. This will give us the inverse function $f^{-1}(x)$.

Step 4: Replace y with f^{-1}(x)

Finally, we replace the variable $y$ with the inverse function $f^{-1}(x)$.

Finding the Inverse of f(x) = (2x-3)/5

Now that we have a general understanding of how to find the inverse of a function, let's apply this process to the given function $f(x)=\frac{2x-3}{5}$.

Step 1: Replace f(x) with y

$f(x)=\frac{2x-3}{5}$

Step 2: Swap x and y

$x=\frac{2y-3}{5}$

Step 3: Solve for y

To solve for $y$, we need to isolate $y$ on one side of the equation. We can do this by multiplying both sides of the equation by 5, which will eliminate the fraction.

$5x=\frac{2y-3}{5}\cdot5$

$5x=2y-3$

Next, we add 3 to both sides of the equation to get rid of the negative term.

$5x+3=2y$

Finally, we divide both sides of the equation by 2 to solve for $y$.

$y=\frac{5x+3}{2}$

Step 4: Replace y with f^{-1}(x)

$f^{-1}(x)=\frac{5x+3}{2}$

Conclusion

In this article, we have learned how to find the inverse of a function using the given function $f(x)=\frac{2x-3}{5}$ as an example. We have followed a step-by-step process to find the inverse function, which involves replacing the function with a variable, swapping the variables, solving for the variable, and replacing the variable with the inverse function. The inverse function $f^{-1}(x)=\frac{5x+3}{2}$ is the correct answer.

Answer

The correct answer is:

A. $f^{-1}(x)=\frac{5x+3}{2}$

Discussion

This problem requires the student to understand the concept of inverse functions and how to find the inverse of a function. The student needs to be able to follow a step-by-step process to find the inverse function, which involves replacing the function with a variable, swapping the variables, solving for the variable, and replacing the variable with the inverse function. The student also needs to be able to evaluate the inverse function and determine the correct answer.

Tips and Variations

• To make this problem more challenging, the student can be asked to find the inverse of a function with a more complex expression, such as $f(x)=\frac{2x^2-3x+1}{5x-2}$.
• To make this problem easier, the student can be given a function with a simpler expression, such as $f(x)=\frac{x-3}{5}$.
• The student can also be asked to find the inverse of a function with a different type of expression, such as a rational expression or a trigonometric expression.

Real-World Applications

The concept of inverse functions has many real-world applications, such as:

• Inverting a matrix to solve a system of linear equations
• Finding the inverse of a function to model a real-world situation
• Using inverse functions to solve optimization problems

Conclusion

Q: What is the inverse of a function?

A: The inverse of a function is a function that undoes the operation of the original 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 value.

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 the function with a variable, such as $y$.
2. Swap the variables $x$ and $y$.
3. Solve for $y$ in terms of $x$.
4. Replace $y$ with the inverse function $f^{-1}(x)$.

Q: What if the function is a rational expression?

A: If the function is a rational expression, you can follow the same steps as above. However, you may need to simplify the expression and cancel out any common factors.

Q: What if the function is a trigonometric expression?

A: If the function is a trigonometric expression, you can follow the same steps as above. However, you may need to use trigonometric identities to simplify the expression.

Q: How do I know if I have found the correct inverse function?

A: To check if you have found the correct inverse function, you can plug in a value of $x$ into the original function and see if the output is equal to the input. If it is, then you have found the correct inverse function.

Q: What if I get a different answer for the inverse function?

A: If you get a different answer for the inverse function, it may be because you made a mistake in one of the steps. Go back and check your work to make sure that you followed the steps correctly.

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

A: Yes, you can use a calculator to find the inverse of a function. However, you need to make sure that you enter the function correctly and that you use the correct inverse function.

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

A: Some common mistakes to avoid when finding the inverse of a function include:

• Not following the steps correctly
• Not simplifying the expression
• Not canceling out common factors
• Not using trigonometric identities when necessary

Q: How do I apply the concept of inverse functions to real-world problems?

A: You can apply the concept of inverse functions to real-world problems by using the inverse function to model a real-world situation. For example, you can use the inverse of a function to model the relationship between the price of a product and the quantity demanded.

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

A: Some real-world applications of inverse functions include:

• Inverting a matrix to solve a system of linear equations
• Finding the inverse of a function to model a real-world situation
• Using inverse functions to solve optimization problems

Conclusion

In conclusion, finding the inverse of a function is an important concept in mathematics that has many real-world applications. By following a step-by-step process and avoiding common mistakes, you can find the inverse of a function and apply it to real-world problems.