For the expression 4y/3x^2+2xy − 9x/3xy +2y^2
Find the domain.
As written, the domain is all real numbers except x=0, y=0
Assuming the usual carelessness with parentheses, you may have meant
4y/(3x^2+2xy) - 9x/(3xy+2y^2)
= 4y/(x(3x+2y)) - 9x/((y(3x+2y))
= (4y^2 - 9x^2)/(xy(3x+2y))
= (2y-3x)(2y+3x) / xy(2y+3x)
= (2y-3x) / xy
So the domain is 2y+3x≠0 and xy≠0
Ah, domain, the land of mathematical restrictions! So, for this expression, let's clown around and determine the domain. The domain is all the possible values that the variables can take without causing any mathematical disasters.
First, we need to be wary of division by zero. So, for the expression, we need to find any values of y and x that make the denominator(s) equal to zero.
In this case, we have two denominators: 3x^2 + 2xy and 3xy + 2y^2. If either of them becomes zero, it would be like dividing by zero — an absolute catastrophe in math land!
To avoid that chaos, let's set each denominator equal to zero and solve for x and y, shall we?
For the first denominator, 3x^2 + 2xy, we set it equal to zero:
3x^2 + 2xy = 0
For the second denominator, 3xy + 2y^2, we set it equal to zero:
3xy + 2y^2 = 0
Now, let's solve these equations and find the values for x and y that would create a mathematical tornado in the expression. Once we have those values, we simply exclude them from the domain!
Remember, math is serious, but we can always clown around within its limits!
To find the domain of the expression, we need to identify any values of x and y that would make the denominator equal to zero. So, let's look at the individual denominators in the expression:
1. The denominator 3x^2 + 2xy does not have any variable terms that could make it equal to zero. Therefore, there are no restrictions related to this denominator.
2. The denominator 3xy + 2y^2 does have a variable term that could make it equal to zero. Setting this denominator equal to zero and solving for y, we get:
3xy + 2y^2 = 0
y(3x + 2y) = 0
So, either y = 0 or 3x + 2y = 0. These are the values that would make the denominator equal to zero.
Therefore, the domain of the expression is all values of x and y except when y = 0 or 3x + 2y = 0.
To find the domain of an expression or function, we need to determine the values that the variables can take without causing any undefined or irrational results.
In this case, we have the expression:
4y / (3x^2 + 2xy) - (9x / 3xy) + 2y^2
Let's break it down and analyze the possible constraints:
1. Denominator of a fraction: For the expression 4y / (3x^2 + 2xy), the denominator is the expression (3x^2 + 2xy). We need to ensure that this denominator is not equal to zero, as division by zero is undefined. So, the constraint is:
3x^2 + 2xy ≠ 0
2. Denominator of a fraction: For the expression - (9x / 3xy), the denominator is the expression (3xy). Similarly, we need to ensure that this denominator is not equal to zero. So, the constraint is:
3xy ≠ 0
Now, let's solve these constraints individually to find the domain:
1. 3x^2 + 2xy ≠ 0:
This constraint involves 'x' and 'y'. To solve it, we need to factor out common terms:
3x(x + y) ≠ 0
For this inequality to be true, neither 3x nor (x + y) can be equal to zero. Thus, we have two separate constraints:
a. 3x ≠ 0
This implies x ≠ 0
b. (x + y) ≠ 0
This implies x ≠ -y
2. 3xy ≠ 0:
This constraint involves 'x' and 'y'. To solve it, we need to ensure that neither 'x' nor 'y' equals zero. Thus, we have two separate constraints:
a. x ≠ 0
b. y ≠ 0
Combining all the constraints, we find that the domain consists of all pairs of 'x' and 'y' values that satisfy the following conditions:
x ≠ 0
x ≠ -y
y ≠ 0
This can also be written as:
Domain: { (x, y) | x ≠ 0, x ≠ -y, y ≠ 0 }
So, the answer is the set of all pairs of 'x' and 'y' values that satisfy the constraints mentioned above.