Question

the table below displays some of the values for an absolute value function.

x	y
0	-5
1	-4
2	-2
4	-3

use the table to determine the specified key features
domain: < x <
range: y ≤

Explanation:

Step1: Determine the domain

The domain of a function is the set of all possible x - values. From the table, the x - values are 0, 1, 2, and 4. The smallest x - value is 0 and the largest x - value is 4. So the domain is \(0

Step2: Determine the range

The range of a function is the set of all possible y - values. Let's list the y - values from the table: when \(x = 0\), \(y=-5\); when \(x = 1\), \(y=-4\); when \(x = 2\), \(y=-2\); when \(x = 4\), \(y=-3\). We can also recall the general form of an absolute - value function \(y=a|x - h|+k\). The vertex of an absolute - value function is a key point. Let's assume the function is of the form \(y = |x - h|+k\). Let's try to find the vertex. The vertex of an absolute - value function is the point where the function changes direction. Let's check the differences in y - values.

First, let's consider the general shape of an absolute - value function. The vertex is the minimum (if \(a>0\)) or maximum (if \(a < 0\)) point. Let's find the x - value that might be the vertex. Let's assume the function is symmetric around the vertex. Let's check the values:

If we assume the vertex is at \(x = 2\) (since the x - value of 2 has a y - value of - 2, which is higher than the y - values at \(x = 0\) and \(x=4\) in a way that suggests a minimum? Wait, no, if \(y=-2\) is higher than \(y=-5\), \(y = - 4\), \(y=-3\), maybe it's a maximum? Wait, no, absolute - value functions open upwards (if \(a>0\)) with a minimum at the vertex or open downwards (if \(a < 0\)) with a maximum at the vertex.

Wait, let's calculate the differences. Let's suppose the function is \(y=-|x - 2|+k\) (opening downwards). When \(x = 2\), \(y=-|2 - 2|+k=k\). From the table, when \(x = 2\), \(y=-2\), so \(k=-2\). Let's check for \(x = 0\): \(y=-|0 - 2|-2=-2 - 2=-4\)? But in the table, when \(x = 0\), \(y=-5\). Maybe my initial assumption is wrong. Let's just use the given y - values. The y - values are - 5, - 4, - 2, - 3. The maximum y - value among these is - 2. Since it's an absolute - value function, if it's opening downwards (because the y - value at the vertex - like point is higher than the other y - values), the range would be \(y\leq - 2\).

Answer:

Domain: \(0

Range: \(y\leq - 2\)