Finding The Inverse Of F(x) = 8√x A Step By Step Solution
In mathematics, the inverse of a function essentially reverses the operation performed by the original function. If a function f takes an input x and produces an output y, the inverse function, denoted as f⁻¹(x), takes y as an input and produces x as the output. Finding the inverse of a function is a fundamental concept in algebra and calculus, with applications ranging from solving equations to understanding the behavior of mathematical models. This article will delve into the process of finding the inverse of a function, specifically focusing on the function f(x) = 8√x for x ≥ 0. We will explore the steps involved, the underlying principles, and the importance of considering the domain and range when determining the inverse.
The process of finding the inverse of a function involves several key steps, each crucial for arriving at the correct result. First, we replace f(x) with y to simplify the notation and make the equation easier to manipulate. This substitution allows us to work with a standard algebraic form, making the subsequent steps more intuitive. Next, we swap x and y, which is the core step in finding the inverse. This swap reflects the fundamental idea of an inverse function – reversing the roles of input and output. By interchanging x and y, we are essentially setting up the equation to solve for the new y, which will represent the inverse function. After swapping the variables, we solve the resulting equation for y. This involves using algebraic manipulations to isolate y on one side of the equation. The specific steps required will depend on the form of the original function, but common techniques include adding, subtracting, multiplying, dividing, and taking roots or powers. Finally, we replace y with f⁻¹(x) to denote the inverse function explicitly. This notation clearly indicates that we have found the inverse of the original function f(x). In addition to these algebraic steps, it's essential to consider the domain and range of both the original function and its inverse. The domain of the original function becomes the range of the inverse, and vice versa. This consideration is particularly important when dealing with functions that have restricted domains or ranges, as it ensures that the inverse function is properly defined.
Step-by-Step Solution
Let's consider the function:
f(x) = 8√x, for x ≥ 0
Our goal is to find its inverse, f⁻¹(x). We will follow the steps outlined above to arrive at the solution.
Step 1: Replace f(x) with y
First, we replace f(x) with y to make the equation easier to work with:
y = 8√x
This substitution simplifies the notation and sets the stage for the next steps in the process.
Step 2: Swap x and y
Next, we swap x and y. This is the crucial step in finding the inverse function, as it reverses the roles of input and output:
x = 8√y
By interchanging x and y, we are now setting up the equation to solve for the new y, which will represent the inverse function.
Step 3: Solve for y
Now, we need to solve the equation x = 8√y for y. This involves isolating y on one side of the equation. To do this, we first divide both sides by 8:
x / 8 = √y
Next, to eliminate the square root, we square both sides of the equation:
(x / 8)² = (√y)²
This simplifies to:
x² / 64 = y
Step 4: Replace y with f⁻¹(x)
Finally, we replace y with f⁻¹(x) to denote the inverse function explicitly:
f⁻¹(x) = x² / 64
This gives us the inverse function in standard notation. However, we need to consider the domain of the inverse function.
Considering the Domain and Range
The original function, f(x) = 8√x, is defined for x ≥ 0. The range of this function is y ≥ 0, since the square root of a non-negative number is always non-negative, and multiplying by 8 maintains this non-negativity. When finding the inverse, the domain and range swap roles. Therefore, the range of f(x) becomes the domain of f⁻¹(x). This means that the inverse function, f⁻¹(x) = x² / 64, is defined for x ≥ 0.
Conclusion
Therefore, the inverse of the function f(x) = 8√x, for x ≥ 0, is:
f⁻¹(x) = x² / 64, for x ≥ 0
Comparing this result with the given options:
A. f⁻¹(x) = 8x², for x ≥ 0 B. f⁻¹(x) = 64x², for x ≥ 0 C. f⁻¹(x) = x² / 64, for x ≥ 0
We can see that option C is the correct answer.
Finding the inverse of a function is a crucial skill in mathematics, with applications in various fields. The process involves swapping the input and output variables and solving for the new output. It's essential to consider the domain and range of both the original function and its inverse to ensure the inverse function is properly defined. In the case of f(x) = 8√x, the inverse function is f⁻¹(x) = x² / 64*, defined for x ≥ 0. Understanding these concepts allows for a deeper comprehension of mathematical functions and their properties.
To find the inverse of a function like f(x) = 8√x, it's crucial to understand the fundamental principles behind inverse functions. An inverse function, denoted as f⁻¹(x), essentially undoes what the original function f(x) does. In simpler terms, if f(a) = b, then f⁻¹(b) = a. This reversal of operations is the core concept behind finding and working with inverse functions. The process involves several steps, each designed to systematically reverse the operations performed by the original function. These steps include replacing f(x) with y, swapping x and y, solving for y, and finally, replacing y with f⁻¹(x). While these steps might seem straightforward, a thorough understanding of each step and the underlying principles is essential for accurately finding the inverse function.
The first step in finding the inverse is to replace f(x) with y. This is a notational convenience that simplifies the algebraic manipulations in subsequent steps. By replacing f(x) with y, we transform the function notation into a standard algebraic equation, which is often easier to work with. For instance, if we have f(x) = 8√x, replacing f(x) with y gives us y = 8√x. This simple substitution sets the stage for the next crucial step: swapping x and y. Swapping x and y is the heart of the inverse function process. This step reflects the fundamental idea that the inverse function reverses the roles of input and output. In the original function, x is the input, and y is the output. By swapping them, we are essentially asking, "What input y would produce the output x in the original function?" This swap transforms the equation from representing y as a function of x to representing x as a function of y. For example, after swapping x and y in the equation y = 8√x, we get x = 8√y. This new equation is the key to finding the inverse function.
After swapping x and y, the next step is to solve the resulting equation for y. This involves using algebraic techniques to isolate y on one side of the equation. The specific techniques used will depend on the form of the equation, but common methods include adding, subtracting, multiplying, dividing, squaring, taking square roots, and so on. The goal is to undo the operations that are being performed on y until y is by itself. For example, in the equation x = 8√y, we would first divide both sides by 8 to get x / 8 = √y. Then, we would square both sides to eliminate the square root, resulting in (x / 8)² = y, which simplifies to x² / 64 = y. This process of solving for y is where algebraic skills are most critical, as it requires careful manipulation of the equation to isolate y. Once y is isolated, we have an expression for y in terms of x, which represents the inverse function. The final step in finding the inverse function is to replace y with f⁻¹(x). This is a notational step that explicitly indicates that we have found the inverse function. The notation f⁻¹(x) is universally recognized as the inverse of f(x), and using this notation makes it clear that the function we have found reverses the operation of the original function. For instance, after solving for y and obtaining y = x² / 64, we would replace y with f⁻¹(x) to write the inverse function as f⁻¹(x) = x² / 64. This notation clearly communicates that we have found the inverse function and sets the stage for using it in further mathematical operations and analysis. In addition to these algebraic steps, it's crucial to consider the domain and range of both the original function and its inverse. 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. This is a fundamental property of inverse functions and is essential for understanding their behavior. When finding the inverse, we must ensure that the domain of the inverse function is consistent with the range of the original function and vice versa. This consideration is particularly important when dealing with functions that have restricted domains or ranges, such as square root functions or trigonometric functions. By paying attention to the domain and range, we can ensure that the inverse function is well-defined and behaves as expected.
Detailed Solution for f(x) = 8√x
Let's walk through a detailed example of finding the inverse of the function f(x) = 8√x, where x ≥ 0. This function involves a square root and a multiplication, and finding its inverse will illustrate the steps and considerations discussed above. We will follow the steps systematically, starting with replacing f(x) with y and progressing through swapping variables, solving for y, and denoting the inverse function appropriately. Along the way, we will also consider the domain and range of the original function and how they relate to the inverse function. This example will provide a concrete application of the principles of inverse functions and demonstrate the importance of each step in the process. The initial step is to replace f(x) with y. This is a straightforward substitution that simplifies the notation and allows us to work with a standard algebraic equation. For the function f(x) = 8√x, replacing f(x) with y gives us:
y = 8√x
This equation is now in a form that is easier to manipulate algebraically. The next step is to swap x and y, which is the core of the inverse function process. Swapping the variables reverses the roles of input and output, setting up the equation to be solved for the new y, which will represent the inverse function. Swapping x and y in the equation y = 8√x yields:
x = 8√y
This new equation represents x as a function of y and is the key to finding the inverse function. Now, we need to solve this equation for y. Solving for y involves isolating y on one side of the equation using algebraic techniques. In the equation x = 8√y, we first need to undo the multiplication by 8 and then the square root. The first step is to divide both sides by 8:
x / 8 = √y
Next, to eliminate the square root, we square both sides of the equation:
(x / 8)² = (√y)²
This simplifies to:
x² / 64 = y
Now y is isolated, and we have an expression for y in terms of x. Finally, we replace y with f⁻¹(x) to denote the inverse function explicitly. This step uses the standard notation for inverse functions, making it clear that we have found the function that reverses the operation of the original function. Replacing y with f⁻¹(x) in the equation y = x² / 64 gives us:
f⁻¹(x) = x² / 64
This is the inverse function for f(x) = 8√x. However, we still need to consider the domain and range to ensure the inverse function is properly defined. The original function f(x) = 8√x is defined for x ≥ 0 because we cannot take the square root of a negative number. The range of the original function is also y ≥ 0, since the square root of a non-negative number is non-negative, and multiplying by 8 maintains this non-negativity. When finding the inverse, the domain and range swap roles. Therefore, the range of f(x) becomes the domain of f⁻¹(x). This means that the inverse function, f⁻¹(x) = x² / 64, is defined for x ≥ 0. Considering the domain and range is crucial for ensuring that the inverse function is well-defined and behaves as expected. In this case, the domain restriction x ≥ 0 is essential for the inverse function to be a valid inverse. In summary, the inverse of the function f(x) = 8√x, for x ≥ 0, is:
f⁻¹(x) = x² / 64, for x ≥ 0
This detailed example illustrates the step-by-step process of finding the inverse of a function, including replacing f(x) with y, swapping x and y, solving for y, replacing y with f⁻¹(x), and considering the domain and range. By following these steps systematically, we can accurately find the inverse of a wide range of functions. Understanding the principles behind inverse functions and the importance of each step is essential for success in this process.
Selecting the Correct Answer
Having found the inverse function to be f⁻¹(x) = x² / 64, for x ≥ 0, we can now compare this result with the given options and select the correct answer. The process of finding the inverse function involved several steps, including replacing f(x) with y, swapping x and y, solving for y, and replacing y with f⁻¹(x). We also considered the domain and range of the original function and its inverse to ensure the inverse function is properly defined. Now, with the correct inverse function in hand, we can confidently identify the correct option from the given choices.
The given options are:
A. f⁻¹(x) = 8x², for x ≥ 0 B. f⁻¹(x) = 64x², for x ≥ 0 C. f⁻¹(x) = x² / 64, for x ≥ 0
Comparing these options with our result, f⁻¹(x) = x² / 64, for x ≥ 0, it is clear that option C matches our calculated inverse function. The other options, A and B, have different coefficients and do not represent the correct inverse function. Therefore, the correct answer is C.
Correct Answer: C. f⁻¹(x) = x² / 64, for x ≥ 0
This process of comparing the calculated result with the given options is a crucial step in problem-solving. It allows us to verify our solution and ensure that we have correctly applied the concepts and techniques involved. In this case, by systematically finding the inverse function and then comparing it with the options, we were able to confidently select the correct answer. This approach highlights the importance of accuracy and attention to detail in mathematical problem-solving. In conclusion, finding the inverse of a function involves a systematic process of swapping variables, solving for the new variable, and considering the domain and range. By following these steps carefully, we can accurately determine the inverse function and confidently select the correct answer from a set of options. In the case of f(x) = 8√x, for x ≥ 0, the inverse function is f⁻¹(x) = x² / 64, for x ≥ 0, making option C the correct choice.
