What is the domain and range of the relation (x,y):absolute value x plus y greater than or equal to two
To find the domain and range of the relation given by the inequality |x| + y ≥ 2, we need to determine the set of valid values for x and y.
1. Domain: The domain refers to the set of all possible values for x.
Since there are no restrictions on the value of x in the inequality, the domain is all real numbers, or (-∞, ∞).
2. Range: The range represents the set of all possible values for y.
To determine the range, we need to consider different cases based on the absolute value of x.
Case 1: x ≥ 0
When x is positive or zero, the inequality becomes x + y ≥ 2. Rearranging the inequality, we have y ≥ 2 - x.
Case 2: x < 0
When x is negative, the absolute value |x| becomes -x. So, the inequality becomes -x + y ≥ 2. Rearranging the inequality, we get y ≥ 2 + x.
For both cases, notice that the inequality does not have an upper bound on y. Hence, the range is all real numbers, or (-∞, ∞).
In summary:
- The domain is all real numbers, or (-∞, ∞).
- The range is also all real numbers, or (-∞, ∞).
Domain: all real numbers
Range: all real numbers greater than or equal to 2 or less than or equal to -2.