Explain why {y||y| ¡Ü 1} is the same as [-1,1].
For
{y | |y|≤1}
By definition of the absolute function, |y| = y if y≥0, and
|y| = -y if y<0.
Therefore the possible values of |y|≤1 are [-1,1]
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To understand why the set {y | |y| ≤ 1} is the same as the interval [-1, 1], we need to break it down step by step and examine what the expression represents.
1. First, let's look at |y|. The vertical bars, or absolute value notation, mean that we consider the magnitude of the number inside them. So, |y| represents the absolute value of y, or the distance of y from zero on the number line.
2. The inequality |y| ≤ 1 means that the absolute value of y is less than or equal to 1. In other words, it indicates that the distance of y from zero is either 1 or less than 1.
3. Now, let's consider the set {y | |y| ≤ 1}. This set consists of all the possible values of y that satisfy the inequality |y| ≤ 1. In other words, it includes any value of y for which the distance from zero is 1 or less.
4. Since the absolute value function produces a positive value, the set {y | |y| ≤ 1} can be interpreted as all values of y that are within a distance of 1 unit from zero in both positive and negative directions. This forms an interval centered around zero.
5. The interval [-1, 1] represents all the real numbers between -1 and 1, including both -1 and 1 themselves. This is exactly what the set {y | |y| ≤ 1} is representing.
Therefore, we can conclude that {y | |y| ≤ 1} is the same as the interval [-1, 1].