Question

state the domain
y^2-16/3y+12

Answer
find the domain.
y^2-16/3y+12

Answer
simplify
y^2-16/3y+12

Answer
find the domain
5x-15y/x^2-9y^2
Answer
simplify
5x-15y/x^2-9y^2

Answer
find domain
xy-x+y-y^2/x^2-y^2
Answer
simplify
xy-x+y-y^2/x^2-y^2

Answer
a^2+2ac+c^2/a^2+ac-ax-cx
find the domain
Answered by GPT 3.5
Since there are no restrictions on the value of y that would result in division by zero, the domain of the function is all real numbers except for y = -4.
Answered by GPT 3.5
The domain of the function is all real numbers except for the values of y that would make the denominator equal to zero. In this case, the denominator is 3y + 12. So, we need to find the values of y that would satisfy the equation 3y + 12 = 0.

Solving this equation, we have:
3y = -12
y = -4

Therefore, the domain of the function is all real numbers except for y = -4.
Answered by GPT 3.5
To simplify the expression y^2 - 16 / 3y + 12, we can factor the numerator and denominator separately.

First, let's focus on factoring the numerator:
y^2 - 16 can be factored as (y - 4)(y + 4).

Now, let's focus on factoring the denominator:
3y + 12 can be factored as 3(y + 4).

So, the simplified expression is:
(y - 4)(y + 4) / 3(y + 4)

Notice that (y + 4) appears in both the numerator and denominator. We can cancel out this common factor, resulting in:
(y - 4) / 3

Thus, the simplified expression is (y - 4) / 3.
Answered by GPT 3.5
To find the domain of the expression (5x - 15y) / (x^2 - 9y^2), we need to determine the values of x and y that would cause division by zero or would result in undefined values.

First, let's determine the values of x that would make the denominator, (x^2 - 9y^2), equal to zero:
x^2 - 9y^2 = 0
(x - 3y)(x + 3y) = 0

This equation will be equal to zero when either (x - 3y) = 0 or (x + 3y) = 0.
So, the values of x that would make the denominator equal to zero are x = 3y and x = -3y.

Therefore, the domain of the expression is all real numbers except for x = 3y and x = -3y.
Answered by GPT 3.5
To simplify the expression (5x - 15y) / (x^2 - 9y^2), we can first factor the numerator and denominator.

The numerator can be factored out the common factor of 5: 5(x - 3y).

The denominator is the difference of squares, which can be factored as (x - 3y)(x + 3y).

Now we can cancel out the common factor of (x - 3y) from the numerator and denominator:

(5(x - 3y)) / ((x - 3y)(x + 3y))

The factor of (x - 3y) cancels out:

5 / (x + 3y)

Therefore, the simplified expression is 5 / (x + 3y).
Answered by GPT 3.5
To find the domain of the expression (xy - x + y - y^2) / (x^2 - y^2), we need to determine the values of x and y that would cause division by zero or would result in undefined values.

First, let's determine the values of x and y that would make the denominator, (x^2 - y^2), equal to zero:
x^2 - y^2 = 0
(x - y)(x + y) = 0

This equation will be equal to zero when either (x - y) = 0 or (x + y) = 0.
So, the values of x and y that would make the denominator equal to zero are x = y and x = -y.

Next, we need to consider any values of x and y that would result in undefined values. In this case, the expression (xy - x + y - y^2) / (x^2 - y^2) will be undefined if both the numerator and denominator are equal to zero.

To find the values of x and y that would make both the numerator and denominator zero, we can set each of them to zero:
xy - x + y - y^2 = 0
x(x - 1) + (y - 1)(y + 1) = 0

However, this equation does not provide a specific solution for x and y. Instead, it represents a line and a parabola intersecting. So, there are no values of x and y that would simultaneously make the numerator and denominator equal to zero. Therefore, we do not have any additional restrictions on the domain.

Therefore, the domain of the expression is all real numbers except for x = y, x = -y, and any other values that may cause division by zero or undefined values in the future.
Answered by GPT 3.5
To simplify the expression (xy - x + y - y^2) / (x^2 - y^2), we can factor the numerator and the denominator:

Numerator:
xy - x + y - y^2 can be rewritten as (xy - x) + (y - y^2) = x(y - 1) + y(1 - y)

Denominator:
x^2 - y^2 can be factored as (x - y)(x + y).

Now, we can rewrite the expression as:
[x(y - 1) + y(1 - y)] / [(x - y)(x + y)]

Next, we can factor out a -1 from the numerator to simplify further:
- [x(1 - y) - y(1 - y)] / [(x - y)(x + y)]

We can combine like terms in the numerator:
- [(x - y)(1 - y)] / [(x - y)(x + y)]

Now, we can cancel out the common factor of (x - y) in the numerator and denominator:
- (1 - y) / (x + y)

Therefore, the simplified expression is - (1 - y) / (x + y).
Answered by GPT 3.5
To find the domain of the expression (a^2 + 2ac + c^2) / (a^2 + ac - ax - cx), we need to determine the values of a and c that would cause division by zero or would result in undefined values.

First, let's consider the denominator, (a^2 + ac - ax - cx). We cannot divide by zero, so we need to find the values of a and c that would make the denominator equal to zero:
a^2 + ac - ax - cx = 0

We can factor out a common factor of a from the first two terms and c from the last two terms:
a(a + c) - x(a + c) = 0

We have a common factor of (a + c), so we can factor that out:
(a + c)(a - x) = 0

Now, set each factor equal to zero to find the values of a and c:
a + c = 0 or a - x = 0

Solving the first equation, we have:
a = -c

We do not have any additional restrictions on the values of c.

Therefore, the domain of the expression is all real numbers except for the values of a and c that would make the denominator equal to zero, which are a = -c and a = x.