Write a piecewise function for the absolute value function. Explain your reasoning. You can insert a picture of your handwritten equation or use the equation tool.
f (x)= 1.5 |x|
recall the definition of |x|:
|x| = x if x ≥ 0
|x| = -x if x < 0
so what is the answer
is it F(x)= 1.5x
To write a piecewise function for the absolute value function, we need to define different equations for different ranges of x.
The absolute value function |x| returns the distance of a number x from zero on the number line. It is always positive or zero, regardless of the sign of x.
For f(x) = 1.5|x|, we can write the piecewise function as follows:
1. When x is greater than or equal to zero:
f(x) = 1.5x
2. When x is less than zero:
f(x) = 1.5(-x) or -1.5x
The reasoning behind this is that when x is positive or zero, the absolute value of x is just x, so f(x) = 1.5x. However, when x is negative, the absolute value of x is -x, so f(x) = 1.5(-x) or -1.5x.
Here is a handwritten representation of the piecewise function:
f(x) = {
1.5x, if x ≥ 0
-1.5x, if x < 0
}
You can also represent this using the equation tool as:
f(x) = (1.5x, x ≥ 0) or (-1.5x, x < 0)
Please note that the piecewise function represents a linear function that has a slope of 1.5 for positive x-values and a slope of -1.5 for negative x-values.