Rewrite The Expression In A More Readable Format And Ensure It Makes Sense.Find The Inverse Function { F^ -1} $}$ For The Following Expression ${ \left[\frac{4 {5} - \frac{6}{5^5} - \frac{1}{5+8}\right] }$

Introduction

In mathematics, expressions can often be complex and difficult to understand. One way to make these expressions more readable is to simplify them and rewrite them in a more organized format. In this article, we will focus on rewriting the given expression in a more readable format and then find the inverse function of the expression.

The Given Expression

The given expression is:

$$\left[\frac{4}{5} - \frac{6}{5^5} - \frac{1}{5+8}\right]$$

This expression is a combination of fractions and a single fraction with a denominator that is a sum of two numbers.

Rewriting the Expression

To rewrite the expression in a more readable format, we need to simplify it by combining like terms and performing any necessary calculations.

First, let's simplify the fraction $\frac{1}{5+8}$:

$$\frac{1}{5+8} = \frac{1}{13}$$

Now, let's rewrite the expression using this simplified fraction:

$$\left[\frac{4}{5} - \frac{6}{5^5} - \frac{1}{13}\right]$$

Next, let's simplify the fraction $\frac{6}{5^5}$:

$$\frac{6}{5^5} = \frac{6}{3125}$$

Now, let's rewrite the expression using this simplified fraction:

$$\left[\frac{4}{5} - \frac{6}{3125} - \frac{1}{13}\right]$$

Finding the Inverse Function

To find the inverse function of the expression, we need to swap the x and y variables and then solve for y.

Let's start by letting $y = \left[\frac{4}{5} - \frac{6}{3125} - \frac{1}{13}\right]$.

To find the inverse function, we need to swap the x and y variables, so we get:

$$x = \left[\frac{4}{5} - \frac{6}{3125} - \frac{1}{13}\right]$$

Now, let's solve for y:

$$x = \frac{4}{5} - \frac{6}{3125} - \frac{1}{13}$$

To solve for y, we need to isolate the variable y on one side of the equation.

First, let's add $\frac{6}{3125}$ to both sides of the equation:

$$x + \frac{6}{3125} = \frac{4}{5} - \frac{1}{13}$$

Next, let's add $\frac{1}{13}$ to both sides of the equation:

$$x + \frac{6}{3125} + \frac{1}{13} = \frac{4}{5}$$

Now, let's simplify the left-hand side of the equation:

$$x + \frac{6}{3125} + \frac{1}{13} = \frac{4}{5}$$

$$x + \frac{6}{3125} = \frac{4}{5} - \frac{1}{13}$$

To simplify the right-hand side of the equation, we need to find a common denominator for the fractions.

The least common multiple of 5 and 13 is 65, so we can rewrite the fractions with a denominator of 65:

$$x + \frac{6}{3125} = \frac{4}{5} - \frac{1}{13}$$

$$x + \frac{6}{3125} = \frac{4 \times 13}{5 \times 13} - \frac{1 \times 5}{13 \times 5}$$

$$x + \frac{6}{3125} = \frac{52}{65} - \frac{5}{65}$$

Now, let's simplify the right-hand side of the equation:

$$x + \frac{6}{3125} = \frac{52}{65} - \frac{5}{65}$$

$$x + \frac{6}{3125} = \frac{47}{65}$$

To isolate the variable x, we need to subtract $\frac{6}{3125}$ from both sides of the equation:

$$x = \frac{47}{65} - \frac{6}{3125}$$

Now, let's simplify the right-hand side of the equation:

$$x = \frac{47}{65} - \frac{6}{3125}$$

$$x = \frac{47 \times 49}{65 \times 49} - \frac{6}{3125}$$

$$x = \frac{2303}{3185} - \frac{6}{3125}$$

To simplify the right-hand side of the equation, we need to find a common denominator for the fractions.

The least common multiple of 3185 and 3125 is 994375, so we can rewrite the fractions with a denominator of 994375:

$$x = \frac{2303}{3185} - \frac{6}{3125}$$

$$x = \frac{2303 \times 313}{3185 \times 313} - \frac{6 \times 317}{3125 \times 317}$$

$$x = \frac{722159}{994375} - \frac{1902}{994375}$$

Now, let's simplify the right-hand side of the equation:

$$x = \frac{722159}{994375} - \frac{1902}{994375}$$

$$x = \frac{720257}{994375}$$

Therefore, the inverse function of the expression is:

$$f^{-1}(x) = \frac{720257}{994375}$$

Conclusion

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Q: What is the inverse function of an expression?

A: The inverse function of an expression is a function that undoes the action of the original function. In other words, if we have a function f(x) and its inverse function f^{-1}(x), then f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

Q: Why is finding the inverse function important?

A: Finding the inverse function is important because it allows us to solve equations that involve the original function. For example, if we have an equation of the form f(x) = y, we can use the inverse function to solve for x.

Q: How do I find the inverse function of an expression?

A: To find the inverse function of an expression, we need to follow these steps:

1. Start with the original expression and swap the x and y variables.
2. Solve for y to find the inverse function.

Q: What if the original expression is a complex function?

A: If the original expression is a complex function, we may need to use algebraic manipulations and mathematical techniques to simplify the expression and find the inverse function.

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

A: Yes, you can use a calculator to find the inverse function. However, keep in mind that the calculator may not always give you the exact answer, and you may need to simplify the expression manually.

Q: What if I get stuck while finding the inverse function?

A: If you get stuck while finding the inverse function, try breaking down the problem into smaller steps and using algebraic manipulations to simplify the expression. You can also try using online resources or seeking help from a teacher or tutor.

Q: Can I use the inverse function to solve equations?

A: Yes, you can use the inverse function to solve equations. For example, if we have an equation of the form f(x) = y, we can use the inverse function to solve for x.

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

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

• Swapping the x and y variables incorrectly
• Not solving for y correctly
• Not simplifying the expression correctly
• Not checking the domain and range of the inverse function

Q: How do I check the domain and range of the inverse function?

A: To check the domain and range of the inverse function, we need to make sure that the inverse function is defined for all values of x in the original function's domain. We can do this by checking the sign of the derivative of the inverse function.

Q: What are some real-world applications of finding the inverse function?

A: Finding the inverse function has many real-world applications, including:

• Modeling population growth and decline
• Analyzing economic data
• Solving optimization problems
• Modeling physical systems

Q: Can I use the inverse function to solve optimization problems?

A: Yes, you can use the inverse function to solve optimization problems. For example, if we have a function f(x) that represents the cost of a product, we can use the inverse function to find the optimal value of x that minimizes the cost.

Q: What are some common types of optimization problems?

A: Some common types of optimization problems include:

• Linear programming problems
• Quadratic programming problems
• Nonlinear programming problems

Q: How do I use the inverse function to solve optimization problems?

A: To use the inverse function to solve optimization problems, we need to follow these steps:

1. Define the objective function and the constraints.
2. Find the inverse function of the objective function.
3. Use the inverse function to find the optimal value of x that minimizes the cost.

Q: What are some common mistakes to avoid when using the inverse function to solve optimization problems?

A: Some common mistakes to avoid when using the inverse function to solve optimization problems include:

• Not defining the objective function and constraints correctly
• Not finding the inverse function correctly
• Not using the inverse function correctly to find the optimal value of x

Q: Can I use the inverse function to solve systems of equations?

A: Yes, you can use the inverse function to solve systems of equations. For example, if we have a system of equations of the form f(x) = y and g(x) = z, we can use the inverse function to solve for x.

Q: How do I use the inverse function to solve systems of equations?

A: To use the inverse function to solve systems of equations, we need to follow these steps:

1. Define the system of equations.
2. Find the inverse function of one of the equations.
3. Use the inverse function to solve for x.

Q: What are some common types of systems of equations?

A: Some common types of systems of equations include:

• Linear systems of equations
• Quadratic systems of equations
• Nonlinear systems of equations

Q: How do I use the inverse function to solve linear systems of equations?

A: To use the inverse function to solve linear systems of equations, we need to follow these steps:

1. Define the system of equations.
2. Find the inverse function of one of the equations.
3. Use the inverse function to solve for x.

Q: What are some common mistakes to avoid when using the inverse function to solve systems of equations?

A: Some common mistakes to avoid when using the inverse function to solve systems of equations include:

• Not defining the system of equations correctly
• Not finding the inverse function correctly
• Not using the inverse function correctly to solve for x