Finding The Inverse Of Y=100-x^2 A Step-by-Step Solution

Understanding Inverse Functions is crucial in mathematics, allowing us to reverse the relationship between variables. This article delves into the process of finding the inverse of the equation $y = 100 - x^2$, providing a step-by-step solution and explaining the underlying concepts. We will analyze the given options and determine the correct inverse function, ensuring a clear understanding of the mathematical principles involved. Let's embark on this journey of mathematical exploration, unraveling the mysteries of inverse functions and enhancing your problem-solving skills.

Demystifying Inverse Functions

Before we tackle the specific equation, let's first clarify what an inverse function truly represents. An inverse function essentially undoes the operation performed by the original function. In simpler terms, if a function takes an input x and produces an output y, its inverse takes y as input and returns the original x. This reversal of roles between input and output is the core concept behind inverse functions. To find the inverse, we typically swap the x and y variables and then solve the equation for y. However, it's important to remember that not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning that each input corresponds to a unique output, and vice versa. This one-to-one property ensures that the inverse function is well-defined and doesn't produce multiple outputs for a single input.

When dealing with functions like the one in question, $y = 100 - x^2$, we need to be mindful of the domain and range. The domain represents the set of all possible input values (x), while the range represents the set of all possible output values (y). When finding the inverse, the domain and range swap roles. This means the domain of the original function becomes the range of the inverse, and the range of the original function becomes the domain of the inverse. Understanding these domain and range considerations is crucial for accurately determining the inverse function and its limitations. Furthermore, the graphical representation of a function and its inverse provides a visual understanding of their relationship. The graphs of a function and its inverse are reflections of each other across the line y = x. This symmetry highlights the concept of reversing the input and output values. By grasping these fundamental concepts, we can confidently approach the process of finding the inverse of $y = 100 - x^2$ and similar equations.

Step-by-Step Solution: Finding the Inverse of $y = 100 - x^2$

Now, let's dive into the process of finding the inverse of the given equation, $y = 100 - x^2$. This involves a series of algebraic manipulations that will ultimately lead us to the inverse function. The first crucial step is to swap the variables x and y. This fundamental operation reflects the core concept of inverse functions, where the input and output roles are reversed. By interchanging x and y, we set the stage for solving for the new y, which will represent the inverse function. So, after swapping, our equation becomes $x = 100 - y^2$. This seemingly simple step is the cornerstone of finding the inverse, as it transforms the equation into a form where we can isolate the inverse function.

Next, we need to isolate $y^2$. To achieve this, we can subtract 100 from both sides of the equation, resulting in $x - 100 = -y^2$. This step brings us closer to isolating y by grouping the y term on one side of the equation. However, we still have a negative sign in front of $y^2$, which needs to be addressed. To eliminate this negative sign, we can multiply both sides of the equation by -1, giving us $100 - x = y^2$. Now, the $y^2$ term is isolated with a positive coefficient, making it easier to solve for y. This algebraic manipulation is a key step in the process, as it prepares the equation for the final step of taking the square root. The isolation of $y^2$ is a common technique used in solving for inverses of functions involving squared terms.

Finally, to solve for y, we take the square root of both sides of the equation. This gives us $y = extbf{±} \sqrt{100 - x}$. It's crucial to remember that when taking the square root, we need to consider both the positive and negative roots, as both values, when squared, will result in the same expression. This ± sign is a critical part of the solution and indicates that the inverse function has two possible outputs for a given input. This is because the original function, $y = 100 - x^2$, is not one-to-one over its entire domain. The inclusion of both positive and negative roots reflects the fact that the inverse function is defined for a range of values that correspond to both positive and negative inputs in the original function. This step completes the process of finding the inverse, but it's essential to interpret the result in the context of the original function and its domain and range.

Analyzing the Options

Having derived the inverse function, $y = extbf{±} \sqrt{100 - x}$, we can now compare it to the given options and identify the correct answer. Let's examine each option carefully:

• A. $y = extbf{±} \sqrt{100 - x}$: This option perfectly matches the inverse function we derived. It includes both the positive and negative square roots of $(100 - x)$, accurately representing the inverse relationship.
• B. $y = 10 extbf{±} \sqrt{x}$: This option does not match our derived inverse function. The structure of the equation is different, and it does not correctly represent the inverse relationship of $y = 100 - x^2$.
• C. $y = 100 extbf{±} \sqrt{x}$: This option is also incorrect. It has a similar structure to option B but still does not align with the inverse function we calculated.
• D. $y = extbf{±} \sqrt{x - 100}$: This option is incorrect because the expression under the square root is $(x - 100)$, whereas our derived inverse has $(100 - x)$ under the square root. This difference significantly alters the function and its behavior.

Therefore, after careful analysis, it's clear that option A, $y = extbf{±} \sqrt{100 - x}$, is the correct inverse function. This option accurately reflects the algebraic manipulations we performed and captures the inverse relationship between the variables. The other options deviate from the correct form of the inverse function and do not represent the reversed relationship of the original equation.

Key Takeaways and Conclusion

In conclusion, the inverse of the equation $y = 100 - x^2$ is $y = extbf{±} \sqrt{100 - x}$. This solution was obtained through a systematic process of swapping the variables x and y, isolating the y term, and taking the square root of both sides. The importance of remembering the ± sign when taking the square root was emphasized, as it accounts for both positive and negative roots and accurately represents the inverse relationship.

Throughout this exploration, we've reinforced the core concept of inverse functions: the reversal of input and output. We've also highlighted the significance of algebraic manipulation in solving for inverse functions, including techniques like isolating variables and applying appropriate operations to both sides of the equation. Furthermore, the process of analyzing options and comparing them to our derived solution underscores the importance of careful attention to detail and a thorough understanding of mathematical principles.

By mastering the process of finding inverse functions, you equip yourself with a valuable tool for solving a wide range of mathematical problems. The ability to reverse relationships between variables is crucial in various fields, including calculus, physics, and engineering. This understanding not only enhances your problem-solving skills but also deepens your appreciation for the interconnectedness of mathematical concepts. Continue practicing and exploring different types of functions and their inverses to solidify your understanding and expand your mathematical toolkit.