Exploring The Range Of Step Function H(x) = -2⌊x⌋
Hey guys! Today, we're diving into the fascinating world of step functions, specifically focusing on the function h(x) = -2⌊x⌋. This function might look a bit intimidating at first, but don't worry, we'll break it down step by step (pun intended!). Our main goal is to figure out the range of this function – in other words, what possible y values can we get out of it? So, let's put on our mathematical hats and get started!
Understanding the Floor Function and Step Functions
Before we tackle our specific function, h(x) = -2⌊x⌋, it's crucial to understand the backbone of it: the floor function. The floor function, denoted by ⌊x⌋, is like a mathematical bouncer – it takes any real number x and rounds it down to the nearest integer. Think of it as finding the largest integer that is less than or equal to x. For example, ⌊3.14⌋ = 3, ⌊-2.7⌋ = -3, and ⌊5⌋ = 5. See? It always rounds down, even for negative numbers!
Now that we've got the floor function down, let's talk about step functions in general. Step functions, also known as staircase functions, get their name from the way their graphs look – like a series of steps. They're piecewise constant functions, meaning they have constant values over certain intervals, and then they abruptly jump to a different constant value. The floor function itself is a classic example of a step function. Each "step" occurs at an integer value, and the function's value remains constant between integers.
Step functions pop up in various real-world scenarios. Imagine the cost of parking in a garage – you might pay a fixed amount for the first hour, then another fixed amount for the next hour, and so on. Or think about how postage rates sometimes work, with different prices for different weight ranges. These situations can be modeled using step functions, which makes them incredibly useful in practical applications. Understanding the floor function is the first step in deciphering these applications.
Delving into h(x) = -2⌊x⌋
Okay, now let's zoom in on our specific function: h(x) = -2⌊x⌋. This function takes the floor of x and then multiplies the result by -2. This simple multiplication has a significant impact on the range of the function. To really grasp this, let's consider some examples. If x = 2.3, then ⌊x⌋ = 2, and h(x) = -2 * 2 = -4. If x = -1.8, then ⌊x⌋ = -2, and h(x) = -2 * -2 = 4. Notice how the multiplication by -2 flips the sign and doubles the value.
Let's think about what this means in terms of the output values of h(x). Since the floor function always produces an integer, multiplying it by -2 will always result in an even integer. This is a crucial observation. We'll never get a fraction, a decimal, or an odd number out of this function. The multiplication by -2 is the key here. It ensures that the output is always an even integer. Therefore, to pinpoint the range accurately, we need to realize this fundamental characteristic. This multiplication by -2 is a crucial step in understanding the function's behavior.
Determining the Range of h(x)
So, we've established that the output of h(x) = -2⌊x⌋ is always an even integer. But are we getting all even integers? Let's consider positive, negative, and zero inputs to see if we can paint a complete picture of the range.
If x is a positive integer, say x = n where n is a positive integer, then ⌊x⌋ = n, and h(x) = -2n. This gives us negative even integers. For example, if x = 3, h(x) = -6. If x = 5, h(x) = -10. We're generating negative even integers here.
If x is a negative integer, say x = -n where n is a positive integer, then ⌊x⌋ = -n, and h(x) = -2 * (-n) = 2n. This gives us positive even integers. For example, if x = -2, h(x) = 4. If x = -4, h(x) = 8. We are obtaining the positive even integers as well.
What happens when x is zero? Well, ⌊0⌋ = 0, and h(0) = -2 * 0 = 0. So, zero is also included in the range. Now, let's think about what happens with non-integer values. If x is a non-integer, say 2.5, then ⌊x⌋ = 2, and h(x) = -4. If x is -1.3, then ⌊x⌋ = -2, and h(x) = 4. These non-integer values simply fill in the gaps between the integer values, but they don't introduce any new types of output. Analyzing integer and non-integer inputs is crucial for a comprehensive understanding.
Answering the Question: The Range of h(x)
Based on our exploration, we can confidently say that the range of h(x) = -2⌊x⌋ consists of all even integers. We've shown that we can generate both positive and negative even integers, as well as zero. There are no odd integers, fractions, or decimals in the output. Therefore, among the given options, the correct answer is:
D. all even integers
Let's quickly look at why the other options are incorrect:
- A. all real numbers: This is wrong because we only get even integers, not all real numbers.
- B. all rational numbers: This is also incorrect because even integers are just a small subset of rational numbers.
- C. all negative integers: This is partially correct, as we do get all negative even integers, but we also get positive even integers and zero.
So, option D perfectly captures the essence of the range of our step function. Even integers are the key to understanding this function's range.
The Importance of Understanding Function Ranges
Figuring out the range of a function is a fundamental concept in mathematics. It tells us the set of all possible output values, which is essential for understanding the behavior and limitations of the function. The range helps us in various ways:
- Solving equations: Knowing the range can help us determine if a solution to an equation exists. For example, if we're trying to solve h(x) = 5, we immediately know there's no solution because 5 is not in the range of h(x).
- Graphing functions: The range provides valuable information about the vertical extent of the graph. We know that the graph will only have y-values within the range.
- Real-world applications: In practical situations, the range can represent physical limitations or constraints. For instance, if a function models the height of a ball thrown in the air, the range will tell us the maximum height the ball can reach.
Understanding function ranges also lays the groundwork for more advanced mathematical concepts, such as inverse functions and transformations of functions. Mastering function ranges is a key stepping stone in mathematical proficiency.
Real-World Examples of Step Functions
As we discussed earlier, step functions have numerous real-world applications. Let's explore some more examples to solidify our understanding:
- Taxi fares: Taxi fares often have a base charge for the first portion of the ride, and then an additional charge for each subsequent fraction of a mile. This can be modeled using a step function.
- Shipping costs: Shipping costs frequently depend on the weight of the package, with different price tiers for different weight ranges. A step function can effectively represent this relationship.
- Tax brackets: Tax systems often use tax brackets, where different income levels are taxed at different rates. This is another classic example of a step function in action.
- Hourly wages with overtime: If you get paid an hourly wage for the first 40 hours of work in a week, and then a higher rate for overtime hours, your total earnings can be modeled using a step function.
These examples demonstrate the versatility of step functions in modeling real-world situations where values change in discrete steps rather than continuously. Real-world applications highlight the practical relevance of step functions.
Conclusion
So, there you have it! We've successfully navigated the world of step functions and determined that the range of h(x) = -2⌊x⌋ is all even integers. We broke down the function step by step, considered various examples, and connected the concept to real-world applications. Hopefully, you now have a much clearer understanding of step functions and how to find their ranges. Keep practicing, and you'll be a step function pro in no time! Remember, understanding the range is crucial for truly grasping a function's behavior.
