Inverse Function Of √(x-2)/6 Step-by-Step Solution
In the realm of mathematics, understanding inverse functions is crucial for solving various problems and grasping fundamental concepts. This comprehensive guide delves into the process of finding the inverse of a given function, specifically focusing on the function f(x) = √(x-2)/6. We will meticulously dissect the steps involved, ensuring clarity and precision in our explanation. Our goal is to empower you with the knowledge and skills to confidently tackle similar inverse function problems.
Demystifying Inverse Functions
Before we embark on the journey of finding the inverse of f(x) = √(x-2)/6, let's first establish a solid understanding of what inverse functions are. In essence, an inverse function "undoes" the action of the original function. If a function f takes an input x and produces an output y, then its inverse function, denoted as f⁻¹, takes y as input and returns x. Think of it as a reverse process, where the input and output roles are swapped.
Key Characteristics of Inverse Functions:
- Domain and Range Swap: The domain of the original function becomes the range of the inverse function, and vice versa. This is a fundamental property that stems from the "undoing" nature of inverse functions.
- Reflection over y = x: The graphs of a function and its inverse are reflections of each other across the line y = x. This visual representation provides a powerful way to understand the relationship between a function and its inverse.
- Composition Property: If f⁻¹ is indeed the inverse of f, then composing them in either order results in the identity function. That is, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This property serves as a crucial test to verify whether a function is the true inverse.
Understanding these core characteristics will significantly aid in our quest to find the inverse of f(x) = √(x-2)/6. Now, let's delve into the step-by-step process.
The Quest for the Inverse of f(x) = √(x-2)/6: A Step-by-Step Approach
To find the inverse of a function, we employ a systematic approach that involves swapping the roles of x and y and then solving for y. Let's meticulously apply this method to our function, f(x) = √(x-2)/6.
Step 1: Replace f(x) with y
Our initial function is f(x) = √(x-2)/6. To initiate the inverse-finding process, we replace f(x) with y, yielding the equation:
y = √(x-2)/6
This seemingly simple substitution sets the stage for the subsequent steps.
Step 2: Swap x and y
The heart of finding an inverse function lies in swapping the roles of x and y. This reflects the fundamental concept that the inverse function "undoes" the original function, effectively reversing the input-output relationship. By swapping x and y in our equation, we obtain:
x = √(y-2)/6
This equation now represents the inverse relationship, albeit implicitly. Our next task is to explicitly solve for y to express the inverse function in its standard form.
Step 3: Isolate the Square Root
Our goal now is to isolate the term containing y, which is currently under the square root. To achieve this, we multiply both sides of the equation by 6:
6x = √(y-2)
This maneuver effectively removes the denominator and brings us closer to isolating the square root term.
Step 4: Eliminate the Square Root
The presence of the square root hinders our ability to isolate y. To eliminate the square root, we square both sides of the equation:
(6x)² = (√(y-2))²
This simplifies to:
36x² = y - 2
By squaring both sides, we've successfully removed the square root and paved the way for isolating y.
Step 5: Solve for y
We are now in the final stretch of finding the inverse function. To isolate y, we add 2 to both sides of the equation:
36x² + 2 = y
This equation explicitly expresses y in terms of x, representing the inverse function.
Step 6: Express the Inverse Function
Finally, we replace y with the inverse function notation, f⁻¹(x), to formally express the inverse:
f⁻¹(x) = 36x² + 2
This is the inverse function of f(x) = √(x-2)/6. However, we must also consider the domain of the inverse function.
Determining the Domain of the Inverse Function
Recall that the domain of the original function becomes the range of the inverse function, and vice versa. The original function, f(x) = √(x-2)/6, has a domain of x ≥ 2 because the expression under the square root must be non-negative. The range of the original function is y ≥ 0 because the square root function always produces non-negative values. Therefore, the domain of the inverse function, f⁻¹(x) = 36x² + 2, is x ≥ 0, which corresponds to the range of the original function.
Final Answer:
The inverse function of f(x) = √(x-2)/6 is:
f⁻¹(x) = 36x² + 2, for x ≥ 0
Analyzing the Answer Choices and Identifying the Correct Option
Now that we have derived the inverse function, let's examine the provided answer choices and pinpoint the correct one.
The answer choices are:
- A. f⁻¹(x) = 36x² + 2, for x ≥ 0
- B. f⁻¹(x) = 6x² + 2, for x ≥ 0
- C. f⁻¹(x) = 36
By comparing our derived inverse function, f⁻¹(x) = 36x² + 2, for x ≥ 0, with the answer choices, it becomes evident that option A is the correct answer. Options B and C are incorrect as they do not match the derived inverse function.
Visualizing the Function and its Inverse
To further solidify our understanding, let's visualize the function f(x) = √(x-2)/6 and its inverse f⁻¹(x) = 36x² + 2.
As mentioned earlier, the graphs of a function and its inverse are reflections of each other across the line y = x. If we were to plot the graphs of f(x) and f⁻¹(x), we would observe this reflection symmetry. The graph of f(x) = √(x-2)/6 starts at the point (2, 0) and increases gradually as x increases. The graph of its inverse, f⁻¹(x) = 36x² + 2, starts at the point (0, 2) and increases rapidly as x increases.
This visual representation provides a powerful way to confirm that the derived function is indeed the inverse of the original function.
Practical Applications of Inverse Functions
Inverse functions are not merely abstract mathematical concepts; they have numerous practical applications in various fields. Here are a few examples:
- Cryptography: Inverse functions play a crucial role in encryption and decryption algorithms. Encrypting a message involves applying a function, and decrypting it requires applying the inverse function.
- Computer Graphics: In computer graphics, transformations such as rotations and scaling are represented by functions. Inverse functions are used to undo these transformations and restore the original image.
- Scientific Modeling: In scientific modeling, inverse functions are used to solve for input variables given output variables. For example, if a model predicts the temperature based on time, the inverse function can be used to determine the time at which a specific temperature will be reached.
- Economics: In economics, inverse demand functions are used to determine the quantity demanded at a given price.
These are just a few examples of the wide-ranging applications of inverse functions. Understanding inverse functions is essential for anyone pursuing careers in mathematics, science, engineering, or related fields.
Conclusion: Mastering Inverse Functions
In this comprehensive guide, we have meticulously explored the process of finding the inverse of the function f(x) = √(x-2)/6. We began by demystifying the concept of inverse functions, highlighting their key characteristics and properties. We then embarked on a step-by-step journey to derive the inverse function, carefully explaining each step and the underlying reasoning. We also emphasized the importance of determining the domain of the inverse function and verifying the answer. Furthermore, we visualized the function and its inverse and discussed the practical applications of inverse functions in various fields.
By mastering the concepts and techniques presented in this guide, you will be well-equipped to tackle inverse function problems with confidence and precision. Remember, practice is key to solidifying your understanding. So, challenge yourself with similar problems and continue to explore the fascinating world of mathematics.