This function can be broken up into different cases depending on the value of x:
Case 1: x < -2
When x < -2, all three absolute values will be negative. So we have: F(x) = -(x-2) - x - (x+2) = -x +2 - x - x - 2 = -3x
Case 2: -2 ≤ x < 0
When -2 ≤ x < 0, the absolute values |x| and |x+2| will be negative, while |x-2| will be positive. So we have: F(x) = (x-2) - x - (x+2) = x - 2 - x - x - 2 = -2x - 4
Case 3: 0 ≤ x < 2
When 0 ≤ x < 2, the absolute values |x-2| and |x+2| will be negative, while |x| will be positive. So we have: F(x) = (x-2) + x - (x+2) = x - 2 + x - x - 2 = -2
Case 4: x ≥ 2
When x ≥ 2, all three absolute values will be positive. So we have: F(x) = (x-2) + x + (x+2) = x - 2 + x + x + 2 = 3x
Putting it all together, the function F(x) = |x-2|+|x|+|x+2| can be written as: F(x) = -3x for x < -2 -2x - 4 for -2 ≤ x < 0 -2 for 0 ≤ x < 2 3x for x ≥ 2
This function can be broken up into different cases depending on the value of x:
Case 1: x < -2
When x < -2, all three absolute values will be negative. So we have:
F(x) = -(x-2) - x - (x+2) = -x +2 - x - x - 2 = -3x
Case 2: -2 ≤ x < 0
When -2 ≤ x < 0, the absolute values |x| and |x+2| will be negative, while |x-2| will be positive. So we have:
F(x) = (x-2) - x - (x+2) = x - 2 - x - x - 2 = -2x - 4
Case 3: 0 ≤ x < 2
When 0 ≤ x < 2, the absolute values |x-2| and |x+2| will be negative, while |x| will be positive. So we have:
F(x) = (x-2) + x - (x+2) = x - 2 + x - x - 2 = -2
Case 4: x ≥ 2
When x ≥ 2, all three absolute values will be positive. So we have:
F(x) = (x-2) + x + (x+2) = x - 2 + x + x + 2 = 3x
Putting it all together, the function F(x) = |x-2|+|x|+|x+2| can be written as:
F(x) =
-3x for x < -2
-2x - 4 for -2 ≤ x < 0
-2 for 0 ≤ x < 2
3x for x ≥ 2