Range Of The Step Function H(x) = -2 Floor(x)

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of mathematics, specifically tackling a cool problem about step functions. You know, those functions that look like a staircase? Well, we've got a specific one here, $h(x)$ which is represented by the equation $y = -2\lfloor x \rfloor$. The big question we need to answer is: which phrase best describes the range of this function $h(x)$? We're given four options: A. all real numbers, B. all rational numbers, C. all negative integers, and D. all even integers. Let's break it down and figure out which one is the winner!

Understanding Step Functions and the Floor Function

Before we jump into the specifics of $h(x) = -2\lfloor x \rfloor$, let's get a solid grasp on what a step function is and, more importantly, what the floor function, denoted by $\lfloor x \rfloor$, actually does. A step function is a piecewise constant function, meaning it takes on a constant value over certain intervals. Think of it like a series of horizontal steps. The floor function, $\lfloor x \rfloor$, is a key component in defining many step functions. It's defined as the greatest integer less than or equal to $x$. So, for example, $\lfloor 3.7 \rfloor = 3$, $\lfloor -2.3 \rfloor = -3$, and $\lfloor 5 \rfloor = 5$. Notice how it always rounds down to the nearest whole number. This rounding down behavior is crucial for understanding the output of our function $h(x)$. The inputs for the floor function can be any real number, but the outputs are always integers. This is a fundamental property that will directly impact the range of our step function.

Analyzing the Function $h(x) = -2\lfloor x \rfloor$

Now, let's put the floor function to work in our specific function, $h(x) = -2\lfloor x \rfloor$. Remember, the output of $\lfloor x \rfloor$ is always an integer. Let's call this integer $n$. So, we can rewrite our function as $h(x) = -2n$, where $n$ is any integer ($\dots, -3, -2, -1, 0, 1, 2, 3, \dots$). What happens when we multiply an integer $n$ by $-2$? We get another integer! But not just any integer; we get an even integer. Let's test a few values for $x$ to see what outputs we get for $h(x)$:

• If $x = 2.5$, then $\lfloor x \rfloor = \lfloor 2.5 \rfloor = 2$. So, $h(2.5) = -2 \times 2 = -4$.
• If $x = -1.8$, then $\lfloor x \rfloor = \lfloor -1.8 \rfloor = -2$. So, $h(-1.8) = -2 \times (-2) = 4$.
• If $x = 0.1$, then $\lfloor x \rfloor = \lfloor 0.1 \rfloor = 0$. So, $h(0.1) = -2 \times 0 = 0$.
• If $x = -3$, then $\lfloor x \rfloor = \lfloor -3 \rfloor = -3$. So, $h(-3) = -2 \times (-3) = 6$.

Notice a pattern here? The outputs we're getting are $-4, 4, 0, 6$. These are all even integers. Can we get any other kind of number? Since $\lfloor x \rfloor$ only produces integers, and we are multiplying that integer by $-2$, the result must be an even integer. The set of integers includes positive integers, negative integers, and zero. When we multiply any integer by $-2$, the result will always be an even integer. For instance, if $\lfloor x \rfloor = 3$, $h(x) = -6$. If $\lfloor x \rfloor = -5$, $h(x) = 10$. If $\lfloor x \rfloor = 0$, $h(x) = 0$. Zero is considered an even number because it is divisible by 2. This means that the output of our function $h(x)$ is restricted to the set of all even integers.

Evaluating the Options for the Range

Let's look at the options provided and see why only one fits perfectly:

• A. All real numbers: The range is definitely not all real numbers. For example, we can never get $3.14$ or $\sqrt{2}$ as an output for $h(x)$, because the output must be an integer, and specifically an even integer. So, option A is out.
• B. All rational numbers: A rational number is any number that can be expressed as a fraction $p/q$, where $p$ and $q$ are integers and $q$ is not zero. While all integers (and thus all even integers) are rational numbers, not all rational numbers are even integers. For instance, $1/2$ is a rational number but not an even integer. Since our function only outputs even integers, this option is too broad and incorrect.
• C. All negative integers: We've seen examples like $h(-1.8) = 4$ and $h(0.1) = 0$. The outputs aren't restricted to only negative integers. We can get positive even integers and zero. Therefore, option C is not the best description of the range.
• D. All even integers: This option perfectly matches our findings! We established that since $\lfloor x \rfloor$ always yields an integer, multiplying it by $-2$ will always result in an even integer. Furthermore, for any even integer $k$, we can find an $x$ such that $h(x) = k$. For example, if we want $h(x) = 10$, we need $-2\lfloor x \rfloor = 10$, which means $\lfloor x \rfloor = -5$. We can choose $x = -5.5$, for instance, and $\lfloor -5.5 \rfloor = -6$, which gives $-2(-6)=12$, nope. Let's try again. If we want $h(x)=10$, we need $-2 imes ext{integer} = 10$, so the integer must be $-5$. We can achieve $\lfloor x \rfloor = -5$ by choosing any $x$ such that $-5 gtr x gtr -4$. For example, if $x = -4.5$, $\lfloor x \rfloor = -5$, and $h(-4.5) = -2(-5) = 10$. If we want $h(x) = -8$, we need $-2\lfloor x \rfloor = -8$, so $\lfloor x \rfloor = 4$. We can choose $x=4.2$, and $h(4.2) = -2(4) = -8$. If we want $h(x)=0$, we need $-2\lfloor x \rfloor = 0$, so $\lfloor x \rfloor = 0$. We can choose $x=0.7$, and $h(0.7) = -2(0) = 0$. This confirms that the function can produce any even integer, both positive, negative, and zero. Thus, the range of the function $h(x)$ is all even integers.

Conclusion

So, after carefully analyzing the properties of the floor function and how it interacts with the multiplication by $-2$ in our step function $h(x) = -2\lfloor x \rfloor$, we can confidently conclude that the output values are restricted to a specific set. The floor function $\lfloor x \rfloor$ always returns an integer, regardless of the input real number $x$. When this integer is multiplied by $-2$, the result is always an even integer. This means the function $h(x)$ can only produce even integers, encompassing positive even integers, negative even integers, and zero. We also confirmed that every even integer can be achieved as an output by selecting an appropriate input $x$. Therefore, the phrase that best describes the range of the function $h(x)$ is all even integers. So, the correct option is D, guys! Keep practicing these concepts, and you'll master them in no time. See you in the next one!