30. Find Formulas For The Inverse Of Each Of The Following Rational Functions.(a) $y = \frac{5x}{x-2}$(b) $y = \frac{3x+2}{x+4}$

In mathematics, rational functions are a type of function that can be expressed as the ratio of two polynomials. The inverse of a rational function is a function that undoes the action of the original function. In other words, if we have a rational function $f(x)$, then its inverse function $f^{-1}(x)$ satisfies the property that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. In this article, we will explore how to find the inverse of each of the following rational functions:

(a) $y = \frac{5x}{x-2}$

To find the inverse of the rational function $y = \frac{5x}{x-2}$, we need to follow these steps:

Step 1: Write the function in terms of $y$

We start by writing the function in terms of $y$:

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

Step 2: Swap $x$ and $y$

Next, we swap $x$ and $y$ to get:

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

Step 3: Solve for $y$

Now, we need to solve for $y$. To do this, we can start by multiplying both sides of the equation by $y-2$ to get:

$$x(y-2) = 5y$$

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

$$xy - 2x = 5y$$

Next, we can add $2x$ to both sides of the equation to get:

$$xy = 5y + 2x$$

Now, we can subtract $2x$ from both sides of the equation to get:

$$xy - 2x = 5y$$

Step 4: Factor out $y$

We can factor out $y$ from the left-hand side of the equation to get:

$$y(x - 5) = 2x$$

Step 5: Solve for $y$

Finally, we can solve for $y$ by dividing both sides of the equation by $x-5$ to get:

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

Therefore, the inverse of the rational function $y = \frac{5x}{x-2}$ is $y = \frac{2x}{x-5}$.

(b) $y = \frac{3x+2}{x+4}$

To find the inverse of the rational function $y = \frac{3x+2}{x+4}$, we need to follow these steps:

Step 1: Write the function in terms of $y$

We start by writing the function in terms of $y$:

$$y = \frac{3x+2}{x+4}$$

Step 2: Swap $x$ and $y$

Next, we swap $x$ and $y$ to get:

$$x = \frac{3y+2}{y+4}$$

Step 3: Solve for $y$

Now, we need to solve for $y$. To do this, we can start by multiplying both sides of the equation by $y+4$ to get:

$$x(y+4) = 3y+2$$

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

$$xy + 4x = 3y + 2$$

Next, we can subtract $3y$ from both sides of the equation to get:

$$xy - 3y + 4x = 2$$

Now, we can factor out $y$ from the left-hand side of the equation to get:

$$y(x - 3) = 2 - 4x$$

Step 4: Solve for $y$

Finally, we can solve for $y$ by dividing both sides of the equation by $x-3$ to get:

$$y = \frac{2 - 4x}{x-3}$$

Therefore, the inverse of the rational function $y = \frac{3x+2}{x+4}$ is $y = \frac{2 - 4x}{x-3}$.

Conclusion

In this article, we have explored how to find the inverse of each of the following rational functions:

• $y = \frac{5x}{x-2}$
• $y = \frac{3x+2}{x+4}$

We have followed a series of steps to find the inverse of each function, including writing the function in terms of $y$, swapping $x$ and $y$, solving for $y$, and factoring out $y$. By following these steps, we have been able to find the inverse of each function and express it in terms of $x$.

References

• [1] "Rational Functions" by Math Open Reference
• [2] "Inverse Functions" by Math Is Fun

Keywords

• Rational functions
• Inverse functions
• Algebra
• Mathematics

Note

In the previous article, we explored how to find the inverse of each of the following rational functions:

• $y = \frac{5x}{x-2}$
• $y = \frac{3x+2}{x+4}$

In this article, we will answer some common questions that students may have when working with inverse rational functions.

Q: What is the difference between a rational function and an inverse rational function?

A: A rational function is a type of function that can be expressed as the ratio of two polynomials. An inverse rational function is a function that undoes the action of the original rational function. In other words, if we have a rational function $f(x)$, then its inverse function $f^{-1}(x)$ satisfies the property that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Q: How do I know if a rational function has an inverse?

A: A rational function has an inverse if and only if it is one-to-one, meaning that each value of the function corresponds to exactly one value of the input. In other words, if the function is strictly increasing or strictly decreasing, then it has an inverse.

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

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

• Swapping $x$ and $y$ without checking if the function is one-to-one
• Not checking if the function is strictly increasing or strictly decreasing
• Not factoring out $y$ from the left-hand side of the equation
• Not solving for $y$ correctly

Q: How do I know if the inverse of a rational function is a rational function?

A: The inverse of a rational function is a rational function if and only if the original function is a rational function. In other words, if the original function can be expressed as the ratio of two polynomials, then its inverse can also be expressed as the ratio of two polynomials.

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

A: Yes, you can use a calculator to find the inverse of a rational function. However, it's always a good idea to check your work by hand to make sure that the inverse function is correct.

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

A: Inverse rational functions have many real-world applications, including:

• Modeling population growth and decline
• Modeling the spread of diseases
• Modeling the behavior of electrical circuits
• Modeling the behavior of mechanical systems

Conclusion

In this article, we have answered some common questions that students may have when working with inverse rational functions. We have also provided some tips and tricks for finding the inverse of a rational function, as well as some real-world applications of inverse rational functions.

References

• [1] "Rational Functions" by Math Open Reference
• [2] "Inverse Functions" by Math Is Fun
• [3] "Calculus" by Michael Spivak

Keywords

• Rational functions
• Inverse functions
• Algebra
• Mathematics
• Calculus

Note

The content of this article is for educational purposes only and is not intended to be used as a substitute for professional mathematical advice. If you have any questions or concerns about the material presented in this article, please consult with a qualified mathematician or educator.