Finding The Range Of $y=\sqrt[3]{-x}-3$ Over [-8 8]
Hey guys! Let's dive into a super interesting math problem today. We're going to explore the range of the function $y=\sqrt[3]{-x}-3$ when it's graphed only over a specific domain. This means we're limiting the x values we can use, which in turn affects the possible y values, or the range. Understanding how domains affect ranges is crucial in understanding functions, so let's break it down together!
Understanding the Function and the Domain
First, let's really get what we're dealing with. The function we have is $y=\sqrt[3]{-x}-3$. This might look a bit intimidating at first, but let's take it piece by piece. The core part is the cube root, $\sqrt[3]{-x}$. Remember, a cube root is the value that, when multiplied by itself three times, gives you the number inside the root. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. The interesting twist here is the negative sign inside the cube root, which means we're taking the cube root of the opposite of x. Then, we subtract 3 from the result of the cube root. This subtraction shifts the entire graph of the function downwards by 3 units. Now, let’s talk about why this function is so interesting. The cube root function itself can take any real number as input. You can take the cube root of positive numbers, negative numbers, and even zero! This is different from square roots, where you can't take the square root of a negative number (at least not in the realm of real numbers). Think about it: what number multiplied by itself three times gives you -8? The answer is -2! So, cube roots are pretty versatile. But here's where things get specific: we aren't looking at the function for all possible x values. We have a restricted domain. The domain is given as ${x \mid -8 \leq x \leq 8}$. In plain English, this means we are only considering x values that are between -8 and 8, including -8 and 8 themselves. This is a crucial piece of information because it will directly determine the range of our function. If we allowed x to be any number, the range would be all real numbers as well, since the cube root function can produce any real number output. However, by limiting the x values, we are also limiting the possible y values. So, the big question becomes: what are the smallest and largest y values we can get when we plug in x values between -8 and 8? This is where we start to think about how the function behaves and how the cube root and the subtraction affect the outcome.
Determining the Range by Analyzing Endpoints
Okay, so we know our domain is ${x \mid -8 \leq x \leq 8}$. The best way to figure out the range for this function with this domain is to look at the endpoints of the domain. These endpoints, x = -8 and x = 8, will give us the extreme y values, which will help us define the range. Let's start with x = -8. We plug this value into our function: $y = \sqrt[3]-(-8)} - 3$. Notice the double negative! -(-8) becomes +8. So we have - 3$. The cube root of 8 is 2, so: $y = 2 - 3 = -1$. Great! When x = -8, y = -1. This gives us one potential endpoint for our range. Now, let's see what happens when x = 8: $y = \sqrt[3]-8} - 3$. The cube root of -8 is -2 (since -2 * -2 * -2 = -8), so - 3$ behaves. The cube root function is continuous and increasing. This means that as the input to the cube root increases, the output also increases. However, we have a negative sign inside the cube root, so we're taking the cube root of -x. This flips the behavior. As x increases, -x decreases, and therefore $\sqrt[3]{-x}$ also decreases. The subtraction of 3 just shifts everything down but doesn't change the increasing/decreasing behavior. So, our function is actually decreasing over its domain. This means that the smallest x value in our domain (-8) will give us the largest y value, and the largest x value (8) will give us the smallest y value. We've already calculated these! When x = -8, y = -1, which is our maximum y value. When x = 8, y = -5, which is our minimum y value. Therefore, we can confidently say that the range of the function over this domain is all the y values between -5 and -1, inclusive.
Expressing the Range in Set Notation and Conclusion
We've figured out the range! The y values fall between -5 and -1, including -5 and -1 themselves. Now, let's write this in set notation, which is a fancy way of expressing a set of numbers. The range can be written as ${y \mid -5 \leq y \leq -1}$. This notation reads as
