Exploring F(x) = X/2 + 5 Finding F(2), F⁻¹(x), And F⁻¹(5)

This article delves into the intricacies of the function f(x) = x/2 + 5, providing a step-by-step exploration of its properties and applications. We will address the core questions surrounding this function, including evaluating f(2), determining the inverse function f⁻¹(x), and calculating f⁻¹(5). By the end of this guide, you will have a solid understanding of how to work with linear functions and their inverses. Let's embark on this mathematical journey together!

a) Evaluating f(2): A Step-by-Step Approach

The first question we address is finding the value of the function f(x) when x = 2. This involves a straightforward substitution process. The function is defined as f(x) = x/2 + 5, meaning that for any input x, we divide it by 2 and then add 5 to the result. To find f(2), we simply replace x with 2 in the function's formula.

So, we have f(2) = 2/2 + 5. Following the order of operations, we first perform the division: 2 divided by 2 equals 1. Thus, the expression simplifies to f(2) = 1 + 5. Now, we perform the addition: 1 plus 5 equals 6. Therefore, we conclude that f(2) = 6. This means that when the input to the function f is 2, the output is 6. This can be visualized as the point (2, 6) on the graph of the function. Understanding function evaluation is crucial as it forms the foundation for more advanced mathematical concepts. For instance, in calculus, evaluating functions at specific points is a fundamental step in finding limits and derivatives. In real-world applications, functions are often used to model relationships between variables, and evaluating the function at a particular value allows us to predict the outcome for a given input. For example, if f(x) represented the cost of producing x units of a product, then f(2) would represent the cost of producing 2 units. Furthermore, the concept of function evaluation is closely linked to the idea of function composition, where the output of one function becomes the input of another. Mastering this basic skill is essential for anyone pursuing further studies in mathematics, science, or engineering. It also plays a significant role in various fields like economics, computer science, and data analysis, where mathematical models are used extensively. In summary, finding f(2) is a fundamental exercise that illustrates the core principle of function evaluation, which has far-reaching implications across diverse disciplines.

b) Determining the Inverse Function f⁻¹(x): Unveiling the Reverse Operation

Next, we tackle the challenge of finding the inverse function, denoted as f⁻¹(x). The inverse function essentially reverses the operation performed by the original function. In other words, if f(x) takes an input x and produces an output y, then f⁻¹(x) takes the input y and produces the output x. To find the inverse function, we need to reverse the steps involved in the original function.

The original function is f(x) = x/2 + 5. The operations involved are dividing x by 2 and then adding 5. To reverse these operations, we first subtract 5 and then multiply by 2. Let's follow a systematic approach to derive the inverse function. First, we replace f(x) with y, so we have y = x/2 + 5. Next, we swap x and y, which gives us x = y/2 + 5. Now, our goal is to solve this equation for y. To isolate y, we first subtract 5 from both sides of the equation: x - 5 = y/2. Then, we multiply both sides by 2 to get 2(x - 5) = y. Distributing the 2, we have 2x - 10 = y. Finally, we replace y with f⁻¹(x) to denote the inverse function. Thus, we have found the inverse function: f⁻¹(x) = 2x - 10. This means that for any input x, the inverse function multiplies it by 2 and then subtracts 10. The concept of inverse functions is crucial in mathematics and has numerous applications. For example, if f(x) represents the conversion from Celsius to Fahrenheit, then f⁻¹(x) would represent the conversion from Fahrenheit to Celsius. In cryptography, inverse functions are used in encryption and decryption processes. Understanding inverse functions also provides valuable insights into the properties of the original function. For instance, a function has an inverse if and only if it is one-to-one, meaning that each output corresponds to a unique input. Graphically, the inverse function is a reflection of the original function across the line y = x. In summary, finding the inverse function f⁻¹(x) = 2x - 10 is a significant step in understanding the relationship between a function and its reverse operation, with applications spanning various mathematical and real-world contexts.

c) Calculating f⁻¹(5): Applying the Inverse Function

Having determined the inverse function f⁻¹(x) = 2x - 10, we can now proceed to calculate f⁻¹(5). This involves substituting x = 5 into the inverse function's formula. We have f⁻¹(5) = 2(5) - 10. Following the order of operations, we first perform the multiplication: 2 multiplied by 5 equals 10. Thus, the expression simplifies to f⁻¹(5) = 10 - 10. Now, we perform the subtraction: 10 minus 10 equals 0. Therefore, we conclude that f⁻¹(5) = 0. This result has a significant interpretation in relation to the original function. Since f⁻¹(5) = 0, it means that f(0) = 5. In other words, if we input 0 into the original function f(x) = x/2 + 5, we should obtain an output of 5. Let's verify this: f(0) = 0/2 + 5 = 0 + 5 = 5, which confirms our result. Calculating the inverse function at a specific point allows us to find the input value that corresponds to a particular output value of the original function. This is a powerful tool in various applications. For example, if f(x) represents the cost of producing x units of a product, and we want to find the number of units that can be produced for a cost of $5, we would calculate f⁻¹(5). The result would tell us the number of units that can be produced for that cost. Furthermore, calculating f⁻¹(5) reinforces our understanding of the relationship between a function and its inverse. It demonstrates how the inverse function undoes the operation of the original function. This concept is fundamental in solving equations, where we often use inverse operations to isolate a variable. In summary, calculating f⁻¹(5) = 0 is not just a numerical exercise; it provides valuable insights into the properties of the function and its inverse, and it has practical applications in various fields.

In conclusion, we have thoroughly explored the function f(x) = x/2 + 5. We successfully evaluated f(2), determined the inverse function f⁻¹(x) = 2x - 10, and calculated f⁻¹(5). These exercises have not only provided us with specific answers but also deepened our understanding of functions, inverse functions, and their applications in mathematics and beyond. This knowledge serves as a solid foundation for further exploration of more complex mathematical concepts.