write the following absolute value functions as piecewise functions.
i really not get how to do this
f(x) = |x|
i think im supposed to state as positive and negative but i not even know what that x means. i only guessing and saying that for positive it be a positive x and for negative it be a negative x.
this one is harder
g(x) = |x+1|
is this part of formula y = a|x-p|+1 so that mean that x+1 is negative horizontal translation and we need to make it positive going |x-1|
and for negative it be |x+1|
To write absolute value functions as piecewise functions, you need to consider two cases: one for positive values and one for negative values.
Let's break down the function f(x) = |x| as an example.
1. For positive values of x: When x is positive (i.e., x > 0), the absolute value of x is equal to x itself. So, for positive values of x, f(x) = x.
2. For negative values of x: When x is negative (i.e., x < 0), the absolute value of x is equal to the negation of x. In this case, f(x) = -x.
Combining the two cases, the piecewise function representation of f(x) = |x| is:
f(x) = {
x, if x > 0,
-x, if x < 0.
}
Now let's move on to the function g(x) = |x + 1|.
1. For positive values of x + 1: To simplify the expression, we consider the equation x + 1 = 0. Solving it gives us x = -1. When x is greater than -1 (i.e., x > -1), x + 1 is positive, and the absolute value of x + 1 is equal to x + 1. So, for positive values of x, g(x) = x + 1.
2. For negative values of x + 1: When x is less than -1 (i.e., x < -1), x + 1 is negative. In this case, the absolute value of x + 1 is -(x + 1). So, for negative values of x, g(x) = -(x + 1).
Combining the two cases, the piecewise function representation of g(x) = |x + 1| is:
g(x) = {
x + 1, if x > -1,
-(x + 1), if x < -1.
}
That's how you write absolute value functions as piecewise functions.