find domain
(x-3)/(x-1) - 2/(1-x)
The domain of this expression is all real numbers except for x = 1.
1/(2a-b) + (3a-b)/(b-2a)
find domain
To find the domain of this expression, we need to determine the values of 'a' and 'b' for which the expression is defined.
First, let's consider the denominator of the first fraction: 2a - b. This fraction is defined as long as the denominator is not equal to zero. So, 2a - b ≠ 0.
Next, let's consider the denominator of the second fraction: b - 2a. This fraction is defined as long as the denominator is not equal to zero. So, b - 2a ≠ 0.
Combining these conditions, we have two inequalities: 2a - b ≠ 0 and b - 2a ≠ 0.
Simplifying the first inequality, we get 2a ≠ b.
Rearranging the inequality, we have a ≠ (1/2)b.
Simplifying the second inequality, we get b ≠ 2a.
Therefore, the domain of the expression is all values of 'a' and 'b' except when a = (1/2)b and b = 2a.
(a+b)^2/(-a-b)^2
find domain now.
To find the domain of this expression, we need to determine the values of 'a' and 'b' for which the expression is defined.
The denominator of the fraction is (-a - b)^2. This fraction is defined as long as the denominator is not equal to zero. In other words, we need to find values of 'a' and 'b' that make (-a - b)^2 ≠ 0.
Simplifying the inequality, we get (-a - b) ≠ 0.
Rearranging the inequality, we have a + b ≠ 0.
Therefore, the domain of the expression is all values of 'a' and 'b' except when a = -b.
To find the domain of the given expression, we must consider the values of x for which the expression is defined. The expression is defined as long as there are no division by zero or undefined operations.
For the given expression (x-3)/(x-1) - 2/(1-x), we observe two potential cases for undefined operations:
1. Division by zero:
The expression is defined if the denominators (x-1) and (1-x) are not equal to zero simultaneously. So, we need to find the values of x for which (x-1) ≠ 0 and (1-x) ≠ 0.
Simplifying (x-1) ≠ 0, we get x ≠ 1.
Simplifying (1-x) ≠ 0, we get x ≠ 1.
Therefore, the expression is defined for all values of x except x = 1.
2. The expression being undefined:
The expression is also undefined if (x-1) = (1-x) = 0, since it would result in division by zero.
However, since (x-1) cannot be equal to (1-x), as shown in the previous step, this case is not possible.
In conclusion, the domain of the given expression is all real numbers except x = 1.
To find the domain of the given expression, we need to consider any restrictions on the values of 'x' that could result in undefined or non-existent values.
Let's simplify the expression first:
(x-3)/(x-1) - 2/(1-x)
Since we have a term (1-x) in the denominator, we need to check if this term could result in any issues.
First, let's consider the expression (x-1). For (x-1) to be defined, x must not be equal to 1 since dividing by zero is undefined.
Next, let's consider the expression (1-x). We can rewrite this as -(x-1). This expression will be defined for all values of x except when x is equal to 1 as well.
So, we have two restrictions: x cannot be equal to 1 (from (x-1)), and x cannot be equal to 1 (from -(x-1)).
Thus, the domain of the given expression is all real numbers except x = 1. In interval notation, the domain can be represented as (-∞, 1) U (1, ∞).