state the range of each relation:
A. y=absolute value of 2x-3
B. y=the square root of x+1
To determine the range of each relation, we need to find the set of all possible outputs or values of "y".
A. For the relation y = |2x - 3|:
To find the range, we first need to understand the behavior of the absolute value function. The absolute value of any real number x is always non-negative (greater than or equal to zero). So, the expression |2x - 3| will also be non-negative for any value of x.
To find the minimum value of |2x - 3|, we set the expression inside the absolute value bars equal to zero:
2x - 3 = 0
2x = 3
x = 3/2 or 1.5
This means that the minimum value of |2x - 3| is zero, which occurs when x = 1.5.
Now, let's consider when |2x - 3| may be greater than zero. Since the coefficient of x is positive (2), the expression 2x - 3 will be positive when x > 3/2, and negative when x < 3/2. However, since we are dealing with absolute values, any negative output will be automatically converted to a positive value.
As a result, the range of the relation y = |2x - 3| is any non-negative value, or in interval notation: [0, ∞).
B. For the relation y = √(x + 1):
To find the range, we need to consider the behavior of the square root function. The square root of any non-negative real number x is always a non-negative value, including zero.
Since the expression inside the square root, x + 1, is added to 1, it will always yield a value greater than or equal to 1. Hence, the square root of (x + 1) will always be greater than or equal to the square root of 1.
Therefore, the range of the relation y = √(x + 1) is any non-negative value (greater than or equal to zero), or in interval notation: [0, ∞).