Finding The Inverse Function Of F(x) = -x^3 - 9 A Step-by-Step Guide

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In mathematics, the concept of an inverse function is fundamental. Essentially, an inverse function "undoes" the operation of the original function. If we apply a function and then its inverse, we should end up back where we started. This article will guide you through the process of finding the inverse of the function f(x)=−x3−9f(x) = -x^3 - 9, providing a detailed explanation and step-by-step solution.

Understanding Inverse Functions

Before we dive into the specific problem, let's clarify what an inverse function is. Given a function f(x)f(x), its inverse, denoted as f−1(x)f^{-1}(x), satisfies the following condition:

f−1(f(x))=xf^{-1}(f(x)) = x and f(f−1(x))=xf(f^{-1}(x)) = x

This means that if we input xx into ff, and then input the result into f−1f^{-1}, we get xx back. Similarly, if we input xx into f−1f^{-1}, and then input the result into ff, we also get xx back. Not all functions have inverses; a function must be one-to-one (also known as injective) to have an inverse. A one-to-one function is one where each input corresponds to a unique output, and vice versa.

Steps to Find the Inverse Function

To find the inverse of a function, we typically follow these steps:

  1. Replace f(x)f(x) with yy.
  2. Swap xx and yy.
  3. Solve for yy.
  4. Replace yy with f−1(x)f^{-1}(x).

These steps effectively reverse the roles of input and output, allowing us to express the inverse function.

Solving for the Inverse of f(x)=−x3−9f(x) = -x^3 - 9

Now, let's apply these steps to the given function, f(x)=−x3−9f(x) = -x^3 - 9.

Step 1: Replace f(x)f(x) with yy

First, we rewrite the function as:

y=−x3−9y = -x^3 - 9

This substitution makes the equation easier to manipulate in the subsequent steps.

Step 2: Swap xx and yy

Next, we interchange xx and yy:

x=−y3−9x = -y^3 - 9

This step is crucial because it reflects the fundamental idea of an inverse function – swapping the input and output variables.

Step 3: Solve for yy

Now, we need to isolate yy on one side of the equation. To do this, we'll perform the following algebraic manipulations:

  1. Add 9 to both sides:

    x+9=−y3x + 9 = -y^3

  2. Multiply both sides by -1:

    −x−9=y3-x - 9 = y^3

  3. Take the cube root of both sides:

    −x−93=y\sqrt[3]{-x - 9} = y

This gives us yy in terms of xx, which is what we need for the inverse function.

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

Finally, we replace yy with the inverse function notation f−1(x)f^{-1}(x):

f−1(x)=−x−93f^{-1}(x) = \sqrt[3]{-x - 9}

So, the inverse of the function f(x)=−x3−9f(x) = -x^3 - 9 is f−1(x)=−x−93f^{-1}(x) = \sqrt[3]{-x - 9}.

Analyzing the Options

Now that we have found the inverse function, let's compare it with the given options:

A. f−1(x)=x+93f^{-1}(x) = \sqrt[3]{x + 9} B. f−1(x)=−x−93f^{-1}(x) = \sqrt[3]{-x - 9} C. f−1(x)=−−x+93f^{-1}(x) = -\sqrt[3]{-x + 9} D. f−1(x)=−x−93f^{-1}(x) = -\sqrt[3]{x - 9}

Comparing our solution, f−1(x)=−x−93f^{-1}(x) = \sqrt[3]{-x - 9}, with the options, we can see that option B matches our result.

Detailed Explanation of Why Other Options are Incorrect

It's crucial to understand why the other options are incorrect to solidify the process of finding inverse functions. Let's analyze each incorrect option in detail:

Option A: f−1(x)=x+93f^{-1}(x) = \sqrt[3]{x + 9}

This option is incorrect because it doesn't correctly account for the negative sign in front of the x3x^3 term in the original function. If we were to compose this function with the original, the negative sign would not cancel out appropriately.

To illustrate, let's compose f(f−1(x))f(f^{-1}(x)) using this incorrect inverse:

f(f−1(x))=f(x+93)=−(x+93)3−9=−(x+9)−9=−x−18f(f^{-1}(x)) = f(\sqrt[3]{x + 9}) = -(\sqrt[3]{x + 9})^3 - 9 = -(x + 9) - 9 = -x - 18

As we can see, the result is −x−18-x - 18, not xx, which confirms that this is not the correct inverse.

Option C: f−1(x)=−−x+93f^{-1}(x) = -\sqrt[3]{-x + 9}

This option introduces an additional negative sign outside the cube root, which is not present in the correct inverse. While it does account for the negative sign within the cube root to some extent, it overcorrects by adding an external negative sign. Let's verify its incorrectness through composition:

f(f−1(x))=f(−−x+93)=−(−−x+93)3−9=−(−(−x+9))−9=−x+9−9=−xf(f^{-1}(x)) = f(-\sqrt[3]{-x + 9}) = -(-\sqrt[3]{-x + 9})^3 - 9 = -(-(-x + 9)) - 9 = -x + 9 - 9 = -x

Again, the composition does not yield xx, indicating that this option is incorrect.

Option D: f−1(x)=−x−93f^{-1}(x) = -\sqrt[3]{x - 9}

This option also includes an incorrect negative sign outside the cube root and incorrectly manipulates the terms inside the cube root. It essentially flips the signs inside the cube root incorrectly, leading to a function that, when composed with the original, will not result in xx. Let's demonstrate this:

f(f−1(x))=f(−x−93)=−(−x−93)3−9=−(−(x−9))−9=x−9−9=x−18f(f^{-1}(x)) = f(-\sqrt[3]{x - 9}) = -(-\sqrt[3]{x - 9})^3 - 9 = -(-(x - 9)) - 9 = x - 9 - 9 = x - 18

The composition gives us x−18x - 18, not xx, thus confirming that this option is not the inverse function.

Why Option B is the Correct Inverse

Option B, f−1(x)=−x−93f^{-1}(x) = \sqrt[3]{-x - 9}, is the correct inverse function because it accurately reverses the operations performed by the original function. The negative sign inside the cube root is crucial for canceling out the negative sign in front of the x3x^3 term in f(x)f(x), and the subtraction of 9 is correctly reversed. Let's verify this by composing the functions:

f(f−1(x))=f(−x−93)=−(−x−93)3−9=−(−x−9)−9=x+9−9=xf(f^{-1}(x)) = f(\sqrt[3]{-x - 9}) = -(\sqrt[3]{-x - 9})^3 - 9 = -(-x - 9) - 9 = x + 9 - 9 = x

And,

f−1(f(x))=f−1(−x3−9)=−(−x3−9)−93=x3+9−93=x33=xf^{-1}(f(x)) = f^{-1}(-x^3 - 9) = \sqrt[3]{-(-x^3 - 9) - 9} = \sqrt[3]{x^3 + 9 - 9} = \sqrt[3]{x^3} = x

Both compositions result in xx, confirming that f−1(x)=−x−93f^{-1}(x) = \sqrt[3]{-x - 9} is indeed the inverse of f(x)=−x3−9f(x) = -x^3 - 9.

Key Takeaways

  • To find the inverse of a function, swap xx and yy and solve for yy.
  • Pay close attention to signs and algebraic manipulations.
  • Always verify your result by composing the function and its inverse; the result should be xx.
  • Understanding why incorrect options are wrong is as important as knowing why the correct option is right.

Conclusion

In conclusion, the correct inverse function for f(x)=−x3−9f(x) = -x^3 - 9 is f−1(x)=−x−93f^{-1}(x) = \sqrt[3]{-x - 9}, which corresponds to option B. This article provided a detailed, step-by-step solution and explained the reasoning behind each step. By understanding the process and the common pitfalls, you can confidently find the inverses of various functions. The ability to determine inverse functions is a valuable skill in mathematics, with applications in numerous areas, including calculus and cryptography. Understanding the step-by-step process, common mistakes, and the importance of verification through composition ensures a solid grasp of this essential concept. Mastering inverse functions opens doors to more advanced mathematical concepts and problem-solving techniques. Therefore, careful attention to detail and practice are key to proficiency in this area. This step-by-step guide aims to provide a comprehensive understanding of finding inverse functions, specifically addressing the challenges posed by negative signs and cube roots. By following these steps and understanding the underlying principles, you can confidently tackle similar problems and enhance your mathematical skills. Ultimately, the ability to find and verify inverse functions is a testament to a deep understanding of fundamental mathematical concepts. Remember to always verify your solution by composing the function and its inverse, ensuring that the result is indeed xx. This final check is crucial in confirming the accuracy of your work and solidifying your understanding of inverse functions.

By understanding the principles and applying the steps outlined in this article, you can successfully navigate the process of finding inverse functions and enhance your mathematical proficiency. The correct inverse function elegantly reverses the operations of the original function, providing a valuable tool for solving a variety of mathematical problems. The key is to practice, pay attention to detail, and always verify your results.