Calculating F(2) With The Function F(x) = -(11)^x

Hey math enthusiasts! Today, we're diving into a straightforward problem involving function evaluation. Our mission? To determine the value of f(2) given the function rule f(x) = -(11)^x. This might seem intimidating at first, but trust me, it's a piece of cake. Let's break down this mathematical concept and see how we can easily calculate the answer. This explanation is designed to be super clear, so even if you're new to functions, you'll be able to follow along. So, grab your pencils, and let's get started!

Understanding the Function Rule

First off, let's get comfortable with what a function rule actually is. Think of a function rule as a recipe. You put something in (the input, represented by x), and the function rule tells you exactly what to do with it (in this case, raising 11 to the power of x and then taking the negative of the result) to get the output, which is f(x). Our specific recipe, or function rule, is f(x) = -(11)^x. This rule instructs us to take any value of x, raise 11 to the power of that value, and then apply a negative sign to the result. Understanding this function is crucial before we can even begin evaluating f(2). It dictates the operations we need to perform: exponentiation (raising 11 to the power of something) and negation (applying a negative sign). Now, don't let the negative sign outside the parentheses confuse you. It's simply saying, "Once you've calculated 11 raised to the power of x, make the answer negative." This seemingly simple rule is the key to solving our problem, and it's something we need to fully grasp to ensure our calculations are accurate. We're going to apply this rule to the specific case where x equals 2. The core idea is that we are substituting 2 for every instance of x in the function. We're essentially swapping out the variable x with the number 2 to determine what the function f will output when x is 2.

So, remember, functions are like mathematical machines – you input a value, it processes it according to a defined rule, and then it spits out an output. This output, f(x), depends entirely on the input value x and the rule specified. In our case, the rule is f(x) = -(11)^x. Therefore, if we plug in 2, the function tells us to raise 11 to the power of 2 (11 squared) and then take the negative of that result. The entire process hinges on correctly interpreting the function rule and applying it in the correct order.

Step-by-Step Calculation of f(2)

Alright, guys, let's get down to the nitty-gritty and calculate f(2). We know our function is f(x) = -(11)^x. We want to find f(2), which means we substitute 2 for x in the function. So, we get f(2) = -(11)^2. Now, we need to evaluate this expression step-by-step. First, we need to calculate 11 squared. Remember that 11 squared (11^2) means 11 multiplied by itself: 11 * 11 = 121. Therefore, (11)^2 equals 121. Now, we apply the negative sign. Our equation becomes f(2) = -121. That's it! We've successfully calculated f(2).

So, to recap the steps: we started with f(x) = -(11)^x. We substituted x with 2, which gave us f(2) = -(11)^2. We then evaluated 11^2, which equals 121. Finally, we applied the negative sign, giving us the answer: f(2) = -121. It's important to remember the order of operations here. We first handled the exponentiation (the power of 2) before applying the negation. We didn't negate the 11 before squaring it. That would have changed the entire outcome. The order is extremely important when dealing with this kind of math problem. We're applying the rules of exponents and then using the negative sign to flip the sign of the result. When we apply the f(x) = -(11)^x rule, we're not just saying "multiply by -11 and then square it", as that would be incorrect due to how math works.

Therefore, understanding the correct steps to calculate f(2) is a cornerstone of grasping function evaluation. We've gone from the initial function rule, made substitutions, performed calculations, and ended up with the correct answer. The process, while simple, helps strengthen our understanding of functions and how they transform input values into output values. Therefore, always remember the order of operations, and the overall process becomes significantly more manageable and less intimidating. Each step is essential. We did the exponentiation before applying the negative sign. The result is -121, not 121. So, with a few simple steps, we've accurately found the value of f(2). It just takes a little bit of practice to become a pro.

The Answer and What It Means

So, after all that work, we have our answer: f(2) = -121. Great job, everyone! What does this result actually mean? In the context of our function f(x) = -(11)^x, it means that when we input 2 into the function, the output is -121. The function takes the input, subjects it to a set of operations (raising 11 to the power of the input and then negating the result), and produces -121 as the output. If we were to graph this function, the point (2, -121) would lie on the curve. This value is a specific data point that describes the behavior of the function at the input value x = 2. This concept is fundamental to understanding functions, as it ties input values directly to the corresponding output values that the function generates.

Basically, the output of -121 tells us precisely what the function does when we feed it the value 2. It's a single snapshot of the function's behavior. If we were to calculate f(3), f(4), or any other f(x), we would get a different output value, demonstrating how the output changes as the input changes. The value -121 itself isn't particularly complex, it simply demonstrates how to accurately follow the function rule. In other words, each input value will provide its unique output. Our answer -121 is the result of using a specific function, which in itself is a method to map an input to an output. And in this case, when we used an input of 2, the function, using its specific rules, gave us an output of -121. The value highlights the relationship between input and output, which is the heart of what functions are all about. It is the core of how functions operate, enabling us to model and analyze many real-world phenomena.

In essence, f(2) = -121 tells us exactly what the function does when x is 2. The negative sign and the exponent are the key parts of the transformation, resulting in this specific output. This is a common function concept that you will meet frequently as you continue your studies. It’s all about the transformation of the input through mathematical operations, resulting in the output of the function.