Understanding The Domain And Graphing Y=√-x Exploring Values At Y=1 And Y=3

by Admin 76 views

Hey guys! Today, we're diving into a fun little math problem that involves understanding square roots, domains, and graphs. We're going to break down the expression √-x, figure out when it makes sense, sketch its graph, and then pinpoint the x-values when y equals 1 and 3. Sounds like a plan? Let's get started!

Understanding the Domain of the Expression √-x

So, our main focus here is on understanding the domain. When we talk about the domain of a function or an expression, we're basically asking, "What are all the possible x-values that we can plug in without causing any mathematical mayhem?" In the case of the square root function, the big no-no is taking the square root of a negative number because, in the world of real numbers, it just doesn't work. You can't find a real number that, when multiplied by itself, gives you a negative result.

Now, let's zoom in on our expression: √-x. Notice that sneaky negative sign inside the square root? That means the value under the square root, which is -x, needs to be greater than or equal to zero. We can write this as an inequality: -x ≥ 0. To solve for x, we can multiply both sides by -1. But here's a crucial thing to remember: when you multiply or divide an inequality by a negative number, you have to flip the inequality sign. So, we get x ≤ 0.

What does this mean? It means that our expression √-x only makes sense when x is zero or a negative number. If we plug in a positive number for x, like 2, we get √-2, which is a no-go in the real number system. But if we plug in a negative number, like -4, we get √-(-4) = √4 = 2, which is perfectly fine. Zero is also allowed because √-0 = √0 = 0.

So, in simple terms, the domain of the expression √-x is all x-values less than or equal to zero. We can represent this using interval notation as (-∞, 0]. This is a fundamental concept in understanding functions, guys, so make sure you've got it down!

Graphing the Function y = √-x

Alright, now that we've nailed down the domain, let's move on to graphing the function y = √-x. Visualizing a function through its graph can give us a much better understanding of its behavior. So, how do we go about sketching this particular graph?

The first thing we need to do is create a table of values. We'll pick some x-values from our domain (remember, x ≤ 0) and calculate the corresponding y-values. This will give us a set of points that we can plot on a coordinate plane. Let's choose a few values:

  • If x = 0, then y = √-0 = √0 = 0. So, we have the point (0, 0).
  • If x = -1, then y = √-(-1) = √1 = 1. So, we have the point (-1, 1).
  • If x = -4, then y = √-(-4) = √4 = 2. So, we have the point (-4, 2).
  • If x = -9, then y = √-(-9) = √9 = 3. So, we have the point (-9, 3).

Now, let's plot these points on a graph. You'll notice that they form a curve. This curve is actually a reflection of the standard square root function, y = √x, across the y-axis. This is because of the negative sign inside the square root. It's like looking at the square root function in a mirror!

The graph starts at the point (0, 0) and extends to the left. As x becomes more negative, y increases, but at a decreasing rate. This means the curve gets flatter as we move further to the left. You can imagine it as half of a parabola lying on its side.

Graphing functions like this helps us see the relationship between x and y in a clear and intuitive way. It's a powerful tool in mathematics, and guys, it's something you'll use a lot as you continue your math journey!

Showing Values Graphically at y = 1 and y = 3

Okay, we've got the domain down, and we've sketched the graph. Now, let's tackle the last part of our problem: showing the values graphically when y = 1 and y = 3. This is where our graph really shines. It allows us to visually find the x-values that correspond to these specific y-values.

First, let's think about y = 1. We want to find the x-value on our graph where the y-coordinate is 1. Imagine drawing a horizontal line across the graph at y = 1. This line will intersect our curve at one point. That point represents the x-value we're looking for.

Looking at our table of values, we already know that when x = -1, y = 1. So, graphically, we can see that the point of intersection between the line y = 1 and our curve is (-1, 1). Therefore, when y = 1, x = -1.

Now, let's do the same for y = 3. Draw another horizontal line across the graph, this time at y = 3. Again, this line will intersect our curve at one point. Looking at our table of values, we see that when x = -9, y = 3. So, the point of intersection is (-9, 3). This tells us that when y = 3, x = -9.

Graphically showing these values makes it super clear how the x and y values are connected in our function. It's like having a visual map that guides us to the right answers. Guys, this is a skill that will come in handy in all sorts of math problems!

Conclusion

So, there you have it! We've successfully navigated through this problem, exploring the domain of the expression √-x, graphing the function y = √-x, and finding the x-values when y = 1 and y = 3. We've covered some key concepts in mathematics, including domains, square root functions, and graphing techniques.

Remember, the key to mastering math is to break down problems into smaller, manageable steps, and to visualize the concepts whenever possible. Understanding the domain is crucial because it tells us where our function is defined. Graphing the function gives us a visual representation of the relationship between x and y. And using the graph to find specific values helps us connect the math to the real world.

Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this, guys! If you have any questions, don't hesitate to ask. Now go out there and conquer those math challenges!