For the expression
b^2 −4by/2y^2 −by − 4y/b −2y
Find the domain.
use some parentheses to make clear what you mean
b^2 − 4by/2y^2 −by − 4y/b −2y
the LCD is 2by^2, so that is
[b^2(4by^2) - 4by(b) - by(4by^2) - 4y(4y^2) - 2y(4by^2)]/(2by^2)
= (4b^3y^2 - 4b^2y - 4b^2y^3 - 16y^3 - 8by^3)/(2by^2)
on the other hand,
(b^2 −4by)/(2y^2 −by) − 4y/(b −2y)
= (b^2-4by) / y(2y-b) + 4y/(y-2b)
= (b^2-4by + 4y^2) / y(2y-b)
= (2y-b)^2 / y(2y-b)
= (2y-b)/y
The required domain is all real values of y, except y = 0 and y = \frac{b}{2}.
The given expression is
\frac{b^{2} - 4by}{2y^{2} - by} - \frac{4y}{b - 2y}
Let, x = \frac{b^{2} - 4by}{2y^{2} - by} - \frac{4y}{b - 2y}
Therefore, to find the domain of this function means to find those values of variable y for which the x-value exists.
It is clear that the denominator of the two terms in the expression must not be zero for x to exist.
So, {2y^{2} - by} \neq 0
⇒ y(2y - b) \neq 0
⇒ y \neq 0 , y \neq \frac{b}{2}
Again, b - 2y \neq 0
⇒ y \neq \frac{b}{2}
Therefore, the required domain is all real values of y, except y = 0 and y = \frac{b}{2}. (Answer)
In other words 5050 is the right answer.
thanks
Well, the domain of this expression is all the values of 'y' for which the expression is defined. Let's take a closer look at the expression and figure it out, shall we?
Now, we need to avoid any values of 'y' that result in dividing by zero (since that's a big no-no in mathematics). So, let's check for any denominators that could potentially be zero.
In this expression, we have two denominators: '2y^2' and 'b'. We need to make sure both of these denominators aren't equal to zero. Therefore, the domain of this expression is all real numbers except for 'y = 0' and 'b = 0'.
But hey, if 'y' and 'b' decide to take a coffee break from being zero, then this expression becomes defined for all other values of 'y' and 'b'.
Hope that clears things up! Now go forth and enjoy the wonderful world of non-zero denominators.
To find the domain of the expression b^2 −4by/2y^2 −by − 4y/b −2y, we need to consider any restrictions or limitations on the variables involved.
The expression has several variables: b and y. In order to determine the domain, we need to identify any values of b and y that would make the expression undefined or result in division by zero or negative square roots.
Let's analyze each part of the expression individually:
1) b^2 − 4by:
This part of the expression does not have any restrictions on b or y. The expression is defined for all real numbers.
2) 2y^2:
This part of the expression does not have any restrictions on y. The expression is defined for all real numbers.
3) - by:
This part of the expression does not have any restrictions on b or y. The expression is defined for all real numbers.
4) -4y:
This part of the expression does not have any restrictions on y. The expression is defined for all real numbers.
5) b − 2y:
This part of the expression does not have any restrictions on b or y. The expression is defined for all real numbers.
Now, let's consider the denominator 2y^2 − by − 4y:
Since the denominator involves variables y and b, we need to check if any values of y or b would make the denominator equal to zero. If the denominator is equal to zero, it would create division by zero, which is undefined.
To find the values that make the denominator zero, we can set it equal to zero and solve for y:
2y^2 − by − 4y = 0
This is a quadratic equation in terms of y. We can use factoring, completing the square, or the quadratic formula to solve for y.
After solving the quadratic equation, we will get a set of values for y that make the denominator zero.
Once we have the values of y that make the denominator zero, we need to consider any values of b that would also make the denominator zero. We can substitute these values of y back into the denominator and solve for b.
The domain of the expression will then be all values of b and y that satisfy the conditions mentioned above, i.e., all real numbers except for the values that make the denominator zero.