Rewrite The Expression To Clarify Its Meaning And Ensure It Makes Sense. Here Is A Possible Interpretation:Evaluate The Inverse Function F − 1 F^{-1} F − 1 At The Expression { \left(\frac{4}{5} - \frac{6}{5^5} - \frac{1}{518}\right)$}$.

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Introduction

In mathematics, expressions can sometimes be ambiguous or unclear, making it difficult to understand their intended meaning. In this article, we will focus on rewriting the given expression to clarify its meaning and ensure it makes sense. We will also evaluate the inverse function f−1f^{-1} at the expression.

Understanding the Expression

The given expression is (45−655−1518)\left(\frac{4}{5} - \frac{6}{5^5} - \frac{1}{518}\right). At first glance, this expression may seem straightforward, but upon closer inspection, we can see that it involves fractions and a negative term. To clarify its meaning, we need to rewrite the expression in a more manageable form.

Rewriting the Expression

To rewrite the expression, we can start by simplifying the fractions. We can rewrite 655\frac{6}{5^5} as 63125\frac{6}{3125} and 1518\frac{1}{518} as 1518\frac{1}{518}. The expression now becomes (45−63125−1518)\left(\frac{4}{5} - \frac{6}{3125} - \frac{1}{518}\right).

Next, we can find a common denominator for the fractions. The least common multiple (LCM) of 5, 3125, and 518 is 5. The expression now becomes (4⋅31255⋅3125−63125−1518)\left(\frac{4 \cdot 3125}{5 \cdot 3125} - \frac{6}{3125} - \frac{1}{518}\right).

We can simplify the expression further by combining the fractions. The expression now becomes (4⋅3125−6−1⋅535⋅3125)\left(\frac{4 \cdot 3125 - 6 - 1 \cdot 5^3}{5 \cdot 3125}\right).

Evaluating the Inverse Function

Now that we have rewritten the expression, we can evaluate the inverse function f−1f^{-1} at the expression. The inverse function f−1f^{-1} is defined as f−1(x)=1f(x)f^{-1}(x) = \frac{1}{f(x)}.

To evaluate the inverse function, we need to find the value of f(x)f(x) at the given expression. We can do this by substituting the expression into the function f(x)f(x).

Let's assume that f(x)=45−655−1518f(x) = \frac{4}{5} - \frac{6}{5^5} - \frac{1}{518}. We can substitute this expression into the function f(x)f(x) to get f−1(x)=145−655−1518f^{-1}(x) = \frac{1}{\frac{4}{5} - \frac{6}{5^5} - \frac{1}{518}}.

Simplifying the Inverse Function

To simplify the inverse function, we can start by finding a common denominator for the fractions. The least common multiple (LCM) of 5, 3125, and 518 is 5. The inverse function now becomes 14⋅31255⋅3125−63125−1518\frac{1}{\frac{4 \cdot 3125}{5 \cdot 3125} - \frac{6}{3125} - \frac{1}{518}}.

We can simplify the inverse function further by combining the fractions. The inverse function now becomes 14⋅3125−6−1⋅535⋅3125\frac{1}{\frac{4 \cdot 3125 - 6 - 1 \cdot 5^3}{5 \cdot 3125}}.

Final Answer

The final answer is 14⋅3125−6−1⋅535⋅3125\boxed{\frac{1}{\frac{4 \cdot 3125 - 6 - 1 \cdot 5^3}{5 \cdot 3125}}}.

Conclusion

In this article, we rewrote the given expression to clarify its meaning and ensure it makes sense. We also evaluated the inverse function f−1f^{-1} at the expression. The final answer is 14⋅3125−6−1⋅535⋅3125\boxed{\frac{1}{\frac{4 \cdot 3125 - 6 - 1 \cdot 5^3}{5 \cdot 3125}}}.

References

Additional Resources

Introduction

In our previous article, we rewrote the given expression to clarify its meaning and ensure it makes sense. We also evaluated the inverse function f−1f^{-1} at the expression. In this article, we will answer some frequently asked questions (FAQs) related to evaluating the inverse function.

Q: What is the inverse function?

A: The inverse function f−1f^{-1} is defined as f−1(x)=1f(x)f^{-1}(x) = \frac{1}{f(x)}. It is a function that undoes the action of the original function f(x)f(x).

Q: How do I evaluate the inverse function?

A: To evaluate the inverse function, you need to find the value of f(x)f(x) at the given expression. You can do this by substituting the expression into the function f(x)f(x).

Q: What is the common denominator for the fractions?

A: The least common multiple (LCM) of 5, 3125, and 518 is 5. This is the common denominator for the fractions.

Q: How do I simplify the inverse function?

A: To simplify the inverse function, you can start by finding a common denominator for the fractions. Then, you can combine the fractions and simplify the expression.

Q: What is the final answer?

A: The final answer is 14⋅3125−6−1⋅535⋅3125\boxed{\frac{1}{\frac{4 \cdot 3125 - 6 - 1 \cdot 5^3}{5 \cdot 3125}}}.

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

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

  • Not finding a common denominator for the fractions
  • Not combining the fractions correctly
  • Not simplifying the expression correctly

Q: How can I practice evaluating the inverse function?

A: You can practice evaluating the inverse function by working through examples and exercises. You can also use online resources and tools to help you practice.

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

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

  • Calculating the inverse of a matrix
  • Finding the inverse of a function in calculus
  • Solving systems of equations

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to evaluating the inverse function. We hope that this article has been helpful in clarifying the concept of the inverse function and how to evaluate it.

References

Additional Resources