Finding The Inverse Function Of F(x) = 2x + 3

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In mathematics, particularly in algebra and calculus, the concept of an inverse function is fundamental. An inverse function, denoted as f⁻¹(x), essentially "undoes" the operation of the original function f(x). In simpler terms, if f maps x to y, then f⁻¹ maps y back to x. This reversal of mapping is the core idea behind inverse functions. To understand this better, consider a simple analogy. Imagine a machine that takes an ingredient (x) and processes it to produce a final product (y). The inverse function is like a machine that takes the final product (y) and reverses the process to retrieve the original ingredient (x). Mathematically, this relationship is expressed as f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. These equations highlight the key property of inverse functions: when a function and its inverse are applied sequentially, they cancel each other out, returning the initial input. This property is crucial for solving equations, simplifying expressions, and understanding the behavior of functions. For a function to have an inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto). An injective function ensures that each input maps to a unique output, while a surjective function ensures that every element in the codomain has a corresponding element in the domain. Linear functions, except for horizontal lines, generally satisfy these conditions and have inverses.

To find the inverse of a function, we follow a systematic approach that involves algebraic manipulation. This process ensures we accurately reverse the mapping defined by the original function. Let's break down the process into clear, manageable steps. First, replace f(x) with y. This step is purely notational and helps simplify the subsequent algebraic manipulations. It transforms the function from function notation to a more standard equation form, making it easier to work with. For example, if we have f(x) = 2x + 3, we rewrite it as y = 2x + 3. This substitution is a simple but crucial step in preparing the function for inversion. Second, swap x and y. This is the heart of the inverse function process. By interchanging the roles of the input and output variables, we are essentially reversing the mapping defined by the original function. The new equation now represents the inverse relationship. Continuing with our example, swapping x and y in y = 2x + 3 gives us x = 2y + 3. This equation expresses x in terms of y, which is the inverse relationship we are seeking. Third, solve for y. This step involves isolating y on one side of the equation. We use algebraic techniques, such as addition, subtraction, multiplication, and division, to achieve this. The goal is to express y as a function of x. In our example, to solve x = 2y + 3 for y, we first subtract 3 from both sides to get x - 3 = 2y. Then, we divide both sides by 2 to obtain y = (x - 3) / 2. This equation now expresses y as a function of x. Fourth, replace y with f⁻¹(x). This final step is again notational. We replace y with the inverse function notation f⁻¹(x) to clearly indicate that we have found the inverse function. This provides the final expression for the inverse function. In our example, replacing y with f⁻¹(x) in y = (x - 3) / 2 gives us f⁻¹(x) = (x - 3) / 2. This is the inverse function of f(x) = 2x + 3.

Now, let's apply these steps to the given function, f(x) = 2x + 3. This will demonstrate how to systematically find the inverse of a linear function. We will follow the steps outlined earlier to ensure accuracy and clarity. First, replace f(x) with y. This gives us y = 2x + 3. This substitution simplifies the equation and prepares it for the next step in the process. Second, swap x and y. This step is crucial for reversing the mapping defined by the original function. Interchanging x and y in y = 2x + 3 yields x = 2y + 3. This new equation represents the inverse relationship, where x is expressed in terms of y. Third, solve for y. To isolate y, we first subtract 3 from both sides of the equation x = 2y + 3, resulting in x - 3 = 2y. Then, we divide both sides by 2 to obtain y = (x - 3) / 2. This equation now expresses y as a function of x, which is the inverse function we are seeking. Fourth, replace y with f⁻¹(x). This final step provides the formal notation for the inverse function. Replacing y with f⁻¹(x) in y = (x - 3) / 2 gives us f⁻¹(x) = (x - 3) / 2. Thus, the inverse function of f(x) = 2x + 3 is f⁻¹(x) = (x - 3) / 2. This result is crucial for understanding the reverse mapping of the original function. We can further simplify f⁻¹(x) = (x - 3) / 2 to f⁻¹(x) = (1/2)x - 3/2 by distributing the division by 2. This form is often preferred for its clarity and ease of comparison with other functions. The final inverse function, f⁻¹(x) = (1/2)x - 3/2, demonstrates how the original function's operations are reversed. The multiplication by 2 in f(x) becomes division by 2 in f⁻¹(x), and the addition of 3 in f(x) becomes subtraction of 3/2 in f⁻¹(x). This inverse function allows us to find the input x that produces a given output y for the original function.

Now that we have determined the inverse function, let's analyze the provided answer choices to identify the correct one. This involves comparing our derived inverse function with each of the options and selecting the matching expression. By systematically evaluating each choice, we can confidently identify the correct answer. The answer choices given are:

A. f⁻¹(x) = -1/2 x - 3/2 B. f⁻¹(x) = 1/2 x - 3/2 C. f⁻¹(x) = -2x + 3 D. f⁻¹(x) = 2x + 3

Comparing these choices with our derived inverse function, f⁻¹(x) = (1/2)x - 3/2, we can see that option B matches exactly. Option A has the correct constants but the wrong sign for the x term. Option C is not even a fractional slope so that one is incorrect. Option D is the original function and not the inverse. Therefore, the correct answer is B.

To ensure our solution is correct, we can verify the inverse function by checking if f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. This verification process is a crucial step in confirming that the derived function is indeed the inverse. Let's perform these checks. First, we check f⁻¹(f(x)). We have f(x) = 2x + 3 and f⁻¹(x) = (1/2)x - 3/2. So, f⁻¹(f(x)) = f⁻¹(2x + 3) = (1/2)(2x + 3) - 3/2 = x + 3/2 - 3/2 = x. This confirms that the first condition is satisfied. Next, we check f(f⁻¹(x)). We have f⁻¹(x) = (1/2)x - 3/2, so f(f⁻¹(x)) = f((1/2)x - 3/2) = 2((1/2)x - 3/2) + 3 = x - 3 + 3 = x. This confirms that the second condition is also satisfied. Since both f⁻¹(f(x)) = x and f(f⁻¹(x)) = x hold true, we can confidently conclude that f⁻¹(x) = (1/2)x - 3/2 is indeed the inverse function of f(x) = 2x + 3. This verification process highlights the importance of checking inverse functions to ensure accuracy. It provides a mathematical confirmation that the derived function correctly reverses the mapping of the original function.

In conclusion, to find the inverse of the function f(x) = 2x + 3, we followed a systematic approach involving swapping variables and solving for y. This process led us to the inverse function f⁻¹(x) = (1/2)x - 3/2, which corresponds to answer choice B. We further verified our solution by confirming that f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. This comprehensive analysis ensures the accuracy of our result and reinforces the understanding of inverse functions. The ability to find inverse functions is a crucial skill in mathematics, with applications ranging from solving equations to understanding transformations and mappings. By mastering this process, students can enhance their problem-solving abilities and gain a deeper appreciation for the relationships between functions and their inverses. Understanding inverse functions not only helps in solving mathematical problems but also provides a foundational concept for more advanced topics in calculus and analysis. The process of finding and verifying inverse functions reinforces algebraic skills and logical reasoning, which are essential for success in mathematics and related fields. The example discussed here, f(x) = 2x + 3, serves as a clear illustration of how to apply the steps involved in finding the inverse of a linear function. This example can be used as a template for solving similar problems and for developing a strong understanding of inverse functions.