Finding The Inverse Of F(x) = X^2 - 16 Step By Step

Let's dive into the world of inverse functions and tackle the problem of finding the inverse of $f(x) = x^2 - 16$ with the domain restriction $x \geq 0$. Understanding inverse functions is crucial in mathematics, as they essentially "undo" the original function. In simpler terms, 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 produces $x$ as the output. This concept has wide-ranging applications in various fields, including cryptography, computer science, and engineering. When dealing with inverse functions, it is essential to pay close attention to the domain and range, as these can significantly affect the result. The domain of a function is the set of all possible input values ($x$-values), while the range is the set of all possible output values ($y$-values). The domain of the original function becomes the range of the inverse function, and vice versa. This relationship ensures that the inverse function effectively reverses the operation of the original function. In our specific problem, the domain restriction $x \geq 0$ is critical because it ensures that the function $f(x) = x^2 - 16$ has an inverse. Without this restriction, the function would not be one-to-one, meaning it would fail the horizontal line test, and therefore, it wouldn't have a well-defined inverse over its entire domain. When finding the inverse of a function, the first step is to replace $f(x)$ with $y$, which makes the equation easier to manipulate. Then, you swap $x$ and $y$, which is the key step in reversing the roles of input and output. After swapping, you solve the equation for $y$, which gives you the inverse function in terms of $x$. Finally, you express the inverse function using the notation $f^{-1}(x)$ to clearly indicate that it is the inverse of the original function. It's also essential to consider any domain restrictions that may arise during the process, such as avoiding square roots of negative numbers or division by zero. Now, let's apply these steps to the given problem.

Finding the Inverse of $f(x) = x^2 - 16$ with $x \geq 0$

To find the inverse of the function $f(x) = x^2 - 16$ with the domain $x \geq 0$, we follow a systematic approach. The key steps involve swapping the roles of $x$ and $y$ and then solving for $y$. First, we replace $f(x)$ with $y$, which gives us the equation $y = x^2 - 16$. This step is simply a notational change to make the equation easier to work with. Next, we swap $x$ and $y$ to reflect the inverse relationship. This gives us $x = y^2 - 16$. This is the crucial step where we reverse the roles of input and output. Now, we need to solve this equation for $y$. To isolate $y^2$, we add 16 to both sides of the equation, resulting in $x + 16 = y^2$. To solve for $y$, we take the square root of both sides. This gives us $y = \pm \sqrt{x + 16}$. However, since the domain of the original function $f(x)$ is restricted to $x \geq 0$, we need to consider the range of the inverse function. The range of the inverse function corresponds to the domain of the original function. Therefore, we must choose the positive square root to ensure that the inverse function is consistent with the original domain restriction. So, we have $y = \sqrt{x + 16}$. Finally, we express the inverse function using the notation $f^{-1}(x)$. Thus, the inverse function is $f^{-1}(x) = \sqrt{x + 16}$. We should also consider the domain of the inverse function. The domain of $f^{-1}(x)$ is determined by the range of the original function $f(x)$. Since $f(x) = x^2 - 16$ and $x \geq 0$, the range of $f(x)$ is $y \geq -16$. Therefore, the domain of $f^{-1}(x)$ is $x \geq -16$, which is consistent with the square root function requiring a non-negative argument. By carefully following these steps, we have successfully found the inverse of the given function, taking into account the domain restriction and ensuring that the inverse function is well-defined. Now, let's examine the provided options to see which one matches our result.

Analyzing the Answer Choices

Now that we've derived the inverse function, $f^{-1}(x) = \sqrt{x + 16}$, we need to compare it with the given answer choices to identify the correct one. This step is crucial to ensure that our calculations are accurate and that we understand the nuances of each option. Let's examine each option individually:

A. $f^{-1}(x) = \sqrt{x + 16}$

This option matches our derived inverse function exactly. It states that the inverse function is the square root of $x + 16$. This aligns perfectly with our calculations, where we added 16 to $x$ and then took the square root. The presence of the $+16$ inside the square root is significant, as it represents the shift in the function's graph due to the original function's constant term, $-16$. This option seems promising, but we must still analyze the other options to ensure that it is the only correct answer.

B. $f^{-1}(x) = \sqrt{x} + 4$

This option differs from our result. It suggests that we should take the square root of $x$ first and then add 4. This is not the same as adding 16 to $x$ before taking the square root. The order of operations matters significantly in mathematics, and this option represents a different transformation. To further illustrate this difference, consider an example. If $x = 9$, then $\sqrt{x + 16} = \sqrt{9 + 16} = \sqrt{25} = 5$, while $\sqrt{x} + 4 = \sqrt{9} + 4 = 3 + 4 = 7$. These results are different, indicating that this option is incorrect.

C. $f^{-1}(x) = \sqrt{x - 16}$

This option also deviates from our derived inverse function. It proposes subtracting 16 from $x$ before taking the square root. This is the opposite of what we found, where we needed to add 16 to $x$. The difference in the sign inside the square root is crucial, as it changes the domain of the function and the direction of the shift. This option would represent the inverse of a different function, not the one we are analyzing. For instance, if $x = 25$, then $\sqrt{x + 16} = \sqrt{25 + 16} = \sqrt{41}$, while $\sqrt{x - 16} = \sqrt{25 - 16} = \sqrt{9} = 3$. These values are clearly different, confirming that this option is incorrect.

D. $f^{-1}(x) = \sqrt{x} - 4$

This option is another variation that does not match our result. It suggests taking the square root of $x$ and then subtracting 4. This is distinct from adding 16 to $x$ before taking the square root. The subtraction of 4 outside the square root indicates a vertical shift downwards, which is not part of the inverse function we derived. This option is not the inverse of the given function. To demonstrate, if $x = 25$, then $\sqrt{x + 16} = \sqrt{25 + 16} = \sqrt{41}$, while $\sqrt{x} - 4 = \sqrt{25} - 4 = 5 - 4 = 1$. These results differ, confirming that this option is not the correct inverse function.

By carefully analyzing each option and comparing it with our derived inverse function, we can confidently conclude that only option A matches our result. The other options represent different transformations and do not correctly undo the original function $f(x) = x^2 - 16$ with the domain restriction $x \geq 0$.

Conclusion: The Correct Inverse Function

After a detailed examination and step-by-step derivation, we have successfully identified the inverse function of $f(x) = x^2 - 16$ with the domain restriction $x \geq 0$. Our calculations led us to the inverse function $f^{-1}(x) = \sqrt{x + 16}$. By comparing this result with the provided answer choices, we confirmed that option A, $f^{-1}(x) = \sqrt{x + 16}$, is the correct answer. The other options, B, C, and D, represent different transformations and do not correctly reverse the operation of the original function. Understanding inverse functions is a fundamental concept in mathematics, with applications in various fields. The process of finding an inverse involves swapping the roles of input and output and then solving for the new output. It's also crucial to consider domain restrictions, as they can significantly affect the inverse function. In this case, the domain restriction $x \geq 0$ ensured that the original function had a well-defined inverse. By carefully following the steps and analyzing the answer choices, we were able to confidently determine the correct inverse function. This problem highlights the importance of precision and attention to detail in mathematics. Each step in the process, from swapping variables to solving for the inverse, must be executed accurately. Additionally, understanding the implications of domain restrictions is crucial for ensuring that the inverse function is valid. With a solid grasp of these concepts, you can tackle similar problems with confidence and accuracy. Inverse functions are not just a mathematical concept; they are a tool for understanding relationships between functions and their reversals. This understanding is valuable in various applications, from solving equations to modeling real-world phenomena. By mastering the techniques for finding inverse functions, you equip yourself with a powerful tool for mathematical problem-solving.

Therefore, the correct answer is A. $f^{-1}(x) = \sqrt{x + 16}$