Hello again, I am now working on trying to find the relation of x = 1 - y^2, along with the domain and range. Then written in interval notation. I think I am ok with figuring out the domain and range on a graph but my problem is how to solve to get these x and y coordinates. Can anyone help me figure this out. Thanks for any help.
just pick any value for y, and evaluate x=1-y^2
y x
-3 -8
-2 -3
-1 0
0 1
1 0
2 -3
3 -8
If you can figure the domain and range, that's good work. Not sure where youir problem is. The "this" has many possible antecedents.
Of course, I can help you with that! To find the relation of the equation x = 1 - y^2, we'll need to solve for either x or y. Let's solve for y in terms of x.
Starting with the equation x = 1 - y^2, we can rearrange it to isolate y:
x = 1 - y^2
y^2 = 1 - x
Now, to solve for y, we take the square root of both sides (since y^2 = (sqrt(y))^2):
sqrt(y^2) = sqrt(1 - x)
Since we want to find the relation in terms of x, we can rewrite it as:
y = +/- sqrt(1 - x)
Here, the +/- indicates that there are two solutions for each x-value, positive and negative.
Now, let's analyze the domain and range of this relation:
Domain: The domain refers to all the possible x-values that the relation can have. In this case, since we have y = +/- sqrt(1 - x), the value inside the square root (1 - x) must be non-negative. Therefore, 1 - x >= 0. Solving this inequality, we get x <= 1. So, the domain for this relation is (-∞, 1].
Range: The range refers to all the possible y-values that the relation can have. Since we have y = +/- sqrt(1 - x), the square root term can take any non-negative value, including zero. Therefore, there is no restriction on the y-values. The range for this relation is (-∞, +∞).
To express the domain and range in interval notation:
Domain: (-∞, 1]
Range: (-∞, +∞)
I hope this explanation helps you understand how to find the relation, domain, and range of the equation x = 1 - y^2. Let me know if you need any further clarification!