The Function F ( X ) = 5 X + 9 X − 8 F(x) = \frac{5x + 9}{x - 8} F ( X ) = X − 8 5 X + 9 Is One-to-one.a. Find An Equation For F − 1 ( X F^{-1}(x F − 1 ( X ], The Inverse Function.b. Verify That Your Equation Is Correct By Showing That F ( F − 1 ( X ) ) = X F(f^{-1}(x)) = X F ( F − 1 ( X )) = X And $f^{-1}(f(x)) =

Mar 9, 2025 by ADMIN 318 views

The Function of Inverse: A Comprehensive Analysis

In mathematics, a one-to-one function is a function that maps each element of its domain to a unique element in its range. The function $f(x) = \frac{5x + 9}{x - 8}$ is a one-to-one function, and in this article, we will explore the concept of its inverse function, denoted as $f^{-1}(x)$ .

Understanding the Concept of Inverse Functions

An inverse function is a function that reverses the operation of the original function. In other words, if $f(x)$ is a function that maps $x$ to $y$ , then the inverse function $f^{-1}(x)$ maps $y$ back to $x$ . The inverse function is denoted by $f^{-1}(x)$ and is read as "f inverse of x".

Finding the Equation for $f^{-1}(x)$

To find the equation for $f^{-1}(x)$ , we need to follow these steps:

Replace $f(x)$ with $y$ : Replace $f(x)$ with $y$ in the original equation to get $y = \frac{5x + 9}{x - 8}$ .
Interchange $x$ and $y$ : Interchange $x$ and $y$ in the equation to get $x = \frac{5y + 9}{y - 8}$ .
Solve for $y$ : Solve for $y$ in the equation to get the equation for $f^{-1}(x)$ .

To solve for $y$ , we can start by multiplying both sides of the equation by $(y - 8)$ to get:

$x(y - 8) = 5y + 9$

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

$xy - 8x = 5y + 9$

Next, we can add $8x$ to both sides of the equation to get:

$xy = 5y + 17x$

Subtracting $5y$ from both sides of the equation, we get:

$xy - 5y = 17x$

Factoring out $y$ from the left-hand side of the equation, we get:

$y(x - 5) = 17x$

Dividing both sides of the equation by $(x - 5)$ , we get:

$y = \frac{17x}{x - 5}$

Therefore, the equation for $f^{-1}(x)$ is:

$f^{-1}(x) = \frac{17x}{x - 5}$

Verifying the Equation

To verify that the equation for $f^{-1}(x)$ is correct, we need to show that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ .

Step 1: Verifying $f(f^{-1}(x)) = x$

To verify that $f(f^{-1}(x)) = x$ , we can substitute $f^{-1}(x)$ into the original function $f(x)$ :

$f(f^{-1}(x)) = f\left(\frac{17x}{x - 5}\right)$

Substituting $\frac{17x}{x - 5}$ into the original function, we get:

$f\left(\frac{17x}{x - 5}\right) = \frac{5\left(\frac{17x}{x - 5}\right) + 9}{\left(\frac{17x}{x - 5}\right) - 8}$

Simplifying the expression, we get:

$f\left(\frac{17x}{x - 5}\right) = \frac{\frac{85x}{x - 5} + 9(x - 5)}{\frac{17x}{x - 5} - 8(x - 5)}$

Combining the fractions on the left-hand side of the equation, we get:

$f\left(\frac{17x}{x - 5}\right) = \frac{\frac{85x + 9x - 45}{x - 5}}{\frac{17x - 8x + 40}{x - 5}}$

Simplifying the expression, we get:

$f\left(\frac{17x}{x - 5}\right) = \frac{\frac{94x - 45}{x - 5}}{\frac{9x + 40}{x - 5}}$

Dividing the numerator and denominator by $(x - 5)$ , we get:

$f\left(\frac{17x}{x - 5}\right) = \frac{94x - 45}{9x + 40}$

Simplifying the expression, we get:

$f\left(\frac{17x}{x - 5}\right) = \frac{94x - 45}{9x + 40}$

$f\left(<br/> Q&A: Inverse Functions and One-to-One Functions

In the previous article, we explored the concept of inverse functions and one-to-one functions. In this article, we will answer some frequently asked questions about inverse functions and one-to-one functions.

Q: What is a one-to-one function?

A: A one-to-one function is a function that maps each element of its domain to a unique element in its range. In other words, a one-to-one function is a function that never takes on the same value twice.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of the original function. In other words, if $f(x)$ is a function that maps $x$ to $y$ , then the inverse function $f^{-1}(x)$ maps $y$ back to $x$ .

Q: How do I find the equation for the inverse function?

A: To find the equation for the inverse function, you need to follow these steps:

Replace $f(x)$ with $y$ in the original equation.
Interchange $x$ and $y$ in the equation.
Solve for $y$ in the equation.

Q: What is the difference between a one-to-one function and an inverse function?

A: A one-to-one function is a function that maps each element of its domain to a unique element in its range. An inverse function is a function that reverses the operation of the original function. While all one-to-one functions have inverse functions, not all functions have inverse functions.

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

A: No, a function cannot have an inverse function if it is not one-to-one. This is because the inverse function would not be a function, but rather a relation.

Q: How do I verify that the equation for the inverse function is correct?

A: To verify that the equation for the inverse function is correct, you need to show that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ .

Q: What is the significance of the inverse function?

A: The inverse function is significant because it allows us to solve equations that involve the original function. By using the inverse function, we can find the value of the original function that corresponds to a given value of the inverse function.

Q: Can the inverse function be used to solve equations that involve the original function?

A: Yes, the inverse function can be used to solve equations that involve the original function. By using the inverse function, we can find the value of the original function that corresponds to a given value of the inverse function.

Q: How do I use the inverse function to solve equations that involve the original function?

A: To use the inverse function to solve equations that involve the original function, you need to follow these steps:

Replace the original function with the inverse function in the equation.
Solve for the inverse function in the equation.
Use the inverse function to find the value of the original function.

Q: What are some common applications of inverse functions?

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

Solving equations that involve the original function
Finding the value of the original function that corresponds to a given value of the inverse function
Modeling real-world phenomena that involve inverse relationships
Solving optimization problems that involve inverse functions

Q: Can inverse functions be used to model real-world phenomena?

A: Yes, inverse functions can be used to model real-world phenomena that involve inverse relationships. Some examples include:

The relationship between the distance traveled and the time taken to travel a certain distance
The relationship between the amount of money spent and the number of items purchased
The relationship between the temperature and the amount of heat energy transferred

Q: How do I use inverse functions to model real-world phenomena?

A: To use inverse functions to model real-world phenomena, you need to follow these steps:

Identify the inverse relationship between the variables.
Write an equation that represents the inverse relationship.
Use the inverse function to model the real-world phenomenon.

Q: What are some common mistakes to avoid when working with inverse functions?

A: Some common mistakes to avoid when working with inverse functions include:

Not checking if the original function is one-to-one before finding the inverse function
Not verifying that the equation for the inverse function is correct
Not using the inverse function to solve equations that involve the original function
Not checking if the inverse function is a function before using it to model real-world phenomena.