Finding The Range Of F(x) = -x² + 1 Domain And Range Explained

Hey guys! Today, we're diving into the fascinating world of functions, specifically a quadratic function, and figuring out its range. Don't worry, it's not as intimidating as it sounds! We'll break it down step by step, making it super easy to understand. We're going to tackle the function f(x) = -x² + 1, but before we jump into solving it, let's make sure we're all on the same page about what functions and their ranges actually are. Think of a function like a magical machine: you feed it an input (an x-value), and it spits out an output (a y-value). The range is simply the collection of all possible outputs you can get from that machine. Now, the function we're dealing with is f(x) = -x² + 1, a classic quadratic function. The "x²" part tells us it's a parabola, a U-shaped curve. The negative sign in front of the x² flips the parabola upside down, making it look like a sad face. The "+ 1" shifts the whole parabola upwards by one unit. This is important because the shape and position of the parabola will directly influence the range of the function. But there's a catch! We're not allowed to feed this function just any number. We have a specific set of inputs, called the domain: {x | 1 ≤ x ≤ 4, x is an integer}. This means we can only use whole numbers (integers) between 1 and 4, inclusive. So, our possible x-values are 1, 2, 3, and 4. This limited domain will drastically affect what our range looks like. To find the range, we'll need to plug each of these x-values into our function and see what we get out. This is where the real fun begins!

Step-by-Step Calculation of the Range

Okay, guys, let's get our hands dirty and calculate the range! Remember, our function is f(x) = -x² + 1, and our allowed x-values are 1, 2, 3, and 4. We're going to plug each of these x-values into the function and see what y-values pop out. First up, let's try x = 1. Plugging this into our function, we get: f(1) = -(1)² + 1 = -1 + 1 = 0. So, when x is 1, y is 0. That's one piece of our range! Next, let's see what happens when x = 2: f(2) = -(2)² + 1 = -4 + 1 = -3. Ah, another y-value for our range: -3. Notice that the negative sign in front of the x² is crucial here. It makes the y-values negative, which will significantly impact our final range. Now, let's move on to x = 3: f(3) = -(3)² + 1 = -9 + 1 = -8. Our range is growing! We now have 0, -3, and -8 as potential y-values. See how the squared term is making the negative values larger in magnitude? This is characteristic of a downward-facing parabola. Finally, let's plug in x = 4: f(4) = -(4)² + 1 = -16 + 1 = -15. Wow, -15 is the lowest y-value we've seen so far. This makes sense, as we're moving further away from the vertex (the highest point) of the parabola. So, we've calculated the y-values for all our allowed x-values. We found 0, -3, -8, and -15. Now, all that's left to do is gather these y-values together to form our range. But before we do that, let's take a step back and think about what we've done. We've systematically evaluated the function for each x-value in the domain, and we've seen how the negative sign and the squared term influence the resulting y-values. This process is fundamental to understanding how functions work and how to determine their ranges. Now, let's collect our results and present the final answer!

Determining the Correct Answer from the Options

Alright, guys, we've done the hard work! We've calculated the y-values corresponding to our domain: 0, -3, -8, and -15. Now, the final step is to match these values with the answer options provided. Remember, the range is the set of all possible output values, so we're looking for an option that contains all four of these numbers. Let's take a look at the options:

a. {-17, -10, -5, -2} b. {-15, -8, -3, 0} c. {0, 3, 8, 15} d. {1, 4, 9, 16} e. {2, 5, 10, 17}

Scanning through the options, we can quickly spot the correct one. Option b. {-15, -8, -3, 0} perfectly matches the y-values we calculated! That's it! We've successfully determined the range of the function. But let's not stop there. It's crucial to understand why this is the correct answer. The other options contain values that we didn't get when we plugged in our x-values. For example, option a has -17, but we never got -17 as an output. Similarly, option c has positive numbers, but our function, with its negative x² term, will always produce non-positive values (zero or negative) for the given domain. Options d and e are clearly incorrect as they contain only positive numbers. This process of elimination is a valuable technique for tackling multiple-choice questions. By understanding the function's behavior and the meaning of the range, we can confidently identify the correct answer and rule out the incorrect ones. We've not only solved the problem, but we've also gained a deeper understanding of the underlying concepts. And that, my friends, is the real victory!

Key Takeaways and General Strategies for Finding Ranges

Okay, guys, we've successfully navigated this problem, but let's zoom out and extract some key takeaways and general strategies that you can use for finding the ranges of other functions. Understanding these principles will make you a range-finding pro! First and foremost, always, always consider the domain! The domain is your playground, the set of x-values you're allowed to play with. It heavily influences the range. If the domain is restricted, like in our problem, the range will also be restricted. If the domain is all real numbers, the range might be very different. Secondly, visualize the function's graph if you can. A mental picture of the function's curve can give you a huge head start in understanding its range. For a quadratic function like ours, the parabola's orientation (upward or downward) and its vertex (highest or lowest point) are crucial. In our case, the downward-facing parabola meant that the range would be limited to values less than or equal to the y-coordinate of the vertex. Thirdly, plug in key x-values. This is what we did in our step-by-step calculation. For a limited domain, plug in the endpoints and any critical points (like the vertex). For more complex functions, consider plugging in values that might reveal important behavior, like where the function changes direction or approaches infinity. Fourthly, look for patterns and trends. As we plugged in x-values, we noticed how the squared term and the negative sign affected the y-values. Recognizing these patterns can help you predict the range without having to calculate every single output. Finally, check your answer against the options. This is especially important in multiple-choice questions. Make sure your range makes sense in the context of the function and the given options. Eliminate options that contain values that couldn't possibly be in the range. By mastering these strategies, you'll be well-equipped to tackle a wide range of range-finding challenges. Remember, practice makes perfect, so keep exploring different functions and their domains, and you'll become a range-finding wizard in no time!

Wrapping Up: Mastering Functions and Their Ranges

Alright, guys, we've reached the end of our journey into the range of the function f(x) = -x² + 1. We've dissected the problem, performed the calculations, and extracted valuable insights along the way. Hopefully, you're feeling much more confident about functions and their ranges now! We started by understanding the basic concept of a range: the set of all possible output values. Then, we dove into our specific quadratic function, recognizing its downward-facing parabolic shape and the importance of the restricted domain. We systematically calculated the y-values for each x-value in the domain, and we carefully matched our results with the answer options. Along the way, we highlighted key strategies for finding ranges, such as visualizing the graph, plugging in key values, and looking for patterns. Remember, finding the range isn't just about plugging in numbers; it's about understanding how the function behaves and how the domain influences the output values. It's about seeing the bigger picture and connecting the dots between different concepts. As you continue your mathematical adventures, remember these principles. They'll serve you well not only in finding ranges but also in understanding other aspects of functions and their applications. So, keep practicing, keep exploring, and keep asking questions. The world of functions is vast and fascinating, and there's always something new to discover. And who knows, maybe you'll even invent a new function yourself one day! Until then, keep those mathematical gears turning, and I'll see you in the next problem!