Final answer:
The piecewise function for Y = -1.6|x-5| +8 is defined by two cases: For x ≥ 5, Y = -1.6x +16, and for x < 5, Y = 1.6x +8.
Step-by-step explanation:
To find the piecewise function of the given function Y = -1.6|x-5| +8, we must evaluate the absolute value expression within different ranges of x.
Absolute value functions are defined piecewise because they have different expressions depending on whether the input is positive or negative. The general form of a piecewise function for an absolute value expression |x - a| is:
- For x ≥ a: f(x) = x - a
- For x < a: f(x) = -(x - a)
In this case, since the expression inside the absolute value is x - 5, we'll break it into two cases:
- For x ≥ 5: Y = -1.6(x - 5) + 8
- For x < 5: Y = -1.6(-(x - 5)) + 8 = Y = 1.6(x - 5) + 8
This gives us the piecewise function for the given expression:
- For x ≥ 5: Y = -1.6x + 16
- For x < 5: Y = 1.6x + 8