Graph, Domain & Range: Unraveling X² - 4x²y² - 4 = 0

Hey guys! Let's dive into a fun math problem: figuring out the graph, domain, and range of the equation x² - 4x²y² - 4 = 0. Sounds a bit intimidating, right? Don't worry, we'll break it down step-by-step to make it super clear and easy to understand. We'll start with the basics, rewrite the equation, identify its type, and then gently explore its graph, domain, and range. Buckle up, and let's get started!

Rewriting the Equation & Identifying the Type

First things first, let's rearrange our equation, x² - 4x²y² - 4 = 0, a bit to make it more friendly to work with. Our goal here is to isolate y or to identify the equation type. If we move the constant term and rearrange, we get 4x²y² = x² - 4. Now, we can divide both sides by 4x² (assuming x ≠ 0) to solve for y²: y² = (x² - 4) / (4x²). This is a crucial step! From here, we can extract y. Taking the square root of both sides gives us y = ±√((x² - 4) / (4x²)). This can be simplified to y = ±√(x² - 4) / (2|x|).

So, what does this tell us? Well, it tells us this is not a typical equation of a circle or an ellipse. It involves both positive and negative square roots, and it is a good idea to consider these to identify and plot the equation. It will most likely result in a more complex graph. Because of the ± sign in the equation and the square root, it's clear that for every value of x (within the domain), there will be two corresponding values of y (except where the expression inside the square root is zero, or where x equals zero). Recognizing this is key to understanding the full picture. The absolute value of x also suggests some symmetry around the y-axis. Remember, understanding the type of equation helps in predicting the graph's shape.

Now, let's turn our attention to the domain and range, which are important features of any function or equation. The domain refers to all possible x-values that can be plugged into the equation, while the range refers to all the possible y-values that the equation can produce. It's time to dig into those details!

Determining the Domain

Alright, let's figure out the domain of the equation y = ±√(x² - 4) / (2|x|). Remember, the domain is all the x-values that work in our equation without causing any mathematical mayhem. In this case, we have a few things to consider. First, we cannot take the square root of a negative number, which means the term x² - 4 inside the square root must be greater than or equal to zero. This is one of the main restrictions that can determine the domain of the equation.

So, we need x² - 4 ≥ 0. Let's solve this inequality. We can factor it as (x - 2)(x + 2) ≥ 0. This inequality is true when both factors are positive (i.e., x > 2) or when both factors are negative (i.e., x < -2). But wait, there is more! The denominator of our equation also brings a crucial restriction. We have 2|x| in the denominator, so x cannot be equal to zero because that would lead to division by zero, which is undefined. Combining these two insights, the domain is all x-values such that x ≤ -2 or x ≥ 2, excluding x = 0, which is already excluded due to the square root constraint.

Therefore, the domain of the equation x² - 4x²y² - 4 = 0 is (-∞, -2] ∪ [2, ∞). This means that x can be any value less than or equal to -2, or any value greater than or equal to 2. x cannot be a number between -2 and 2 (excluding 0). It also cannot be equal to 0. Understanding the domain helps to visualize the horizontal extent of the graph. You will start to see that the graph will not exist between -2 and 2, which are important points to know when we start to plot the function.

Determining the Range

Okay, let's tackle the range of the equation y = ±√(x² - 4) / (2|x|). Remember, the range is the set of all possible y-values that the equation can produce. This can often be a bit trickier than determining the domain, but don't worry, we'll break it down. We already know the domain, so we can work with that information to figure out the range. Let's remember the behavior of the equation. Since we have y = ±√(x² - 4) / (2|x|), the ± sign means for every valid x-value (in our domain), we will have two corresponding y-values: one positive and one negative. Also, notice that the expression inside the square root must be non-negative, and the denominator 2|x| is always positive (except when x = 0, which is outside our domain).

Given the structure of the equation, the y-values will always be either positive or negative (or zero, when x = ±2), but never a y value in between. As x moves away from 2 or -2, the absolute value of y decreases, approaching 0. At x = ±2, the value of y is 0. Also, as x tends to infinity, the absolute value of y approaches 1, but never actually reaches 1. Given all of this, we know that y can take values that are less than or equal to -1 or greater than or equal to 1, or y = 0. Therefore, the range is y ≤ -1 and y ≥ 1, as well as y = 0. So, we can say that the range of the equation x² - 4x²y² - 4 = 0 is (-∞, -1] ∪ {0} ∪ [1, ∞). This means that y can be any value less than or equal to -1, equal to 0, or greater than or equal to 1.

Plotting the Graph

Time to paint a picture! Now that we know the domain and range, and we've rearranged the equation, we can start to visualize the graph. The equation y = ±√(x² - 4) / (2|x|) gives us a pretty good idea of what our graph will look like. Since we have the ±, this will be a graph with two parts. Let's think this through. We know the domain is (-∞, -2] ∪ [2, ∞). This means our graph won't exist between x = -2 and x = 2. It will exist on the left side of -2 and on the right side of 2, so the graph will stretch out towards infinity in both directions. The y-values are restricted too, (-∞, -1] ∪ {0} ∪ [1, ∞). This means our graph won't exist between -1 and 1.

So, the graph will consist of two curves. One will be above the line y = 1 and another below y = -1. Because of the absolute value, the graph is symmetric about the y-axis. The curve touches the x-axis at x = 2 and x = -2. As x moves towards infinity, y approaches -1 and 1 as the curve gets closer and closer. In general, it will look like two separate curves, opening out horizontally, each extending from x = -2 and x = 2, and going towards infinity. The curves are always at a distance of 1 unit away from the x-axis and y-axis. Remember, a rough sketch can be super helpful. Just plot a few points within your domain to confirm that the graph looks the way you expected. You can test values, such as x = 2, x = -2, x = 3, and x = -3. When x = ±2, y = 0. When x = ±3, y = ±√5 / 6. The more data you have, the clearer the picture will become. Also, you can use graphing tools like Desmos or Wolfram Alpha to plot the graph and confirm your answers.

Conclusion

Awesome work, everyone! We've successfully navigated the graph, domain, and range of the equation x² - 4x²y² - 4 = 0. We started by rearranging the equation, which showed the nature of the equation. We then identified the domain by considering restrictions imposed by the square root and the denominator, and the range based on the behavior of the equation and the square root. We even discussed what the graph looks like. It is an amazing feeling to turn a complex math problem into something clear and understandable! Hopefully, this helps you to understand how to solve similar problems. Keep practicing and exploring, and you'll become a master of all things math.