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|
|N| = N if N >=0
|N| = -N if N < 0
f(x) = |x| =
x if x >= 0
-x if x < 0
g(x) = |x+1| =
x+1 if x+1 >= 0 (that is, if x >= -1)
-(x+1) if x+1 < 0
thanks very much steve :)
To write absolute value functions as piecewise functions, you need to consider the different cases when the input (x) is positive or negative. Here's how you can approach each function:
1. f(x) = |x|
For this function, you are correct that you need to state it for both positive and negative x values separately.
- For positive values of x, the absolute value of x is just x itself. So, f(x) = x.
- For negative values of x, the absolute value of x is equal to the opposite of x. Therefore, for negative x values, f(x) = -x.
Combining these cases, the piecewise function for f(x) = |x| can be written as:
f(x) = {
x, if x >= 0,
-x, if x < 0
}
2. g(x) = |x + 1|
To write g(x) as a piecewise function, you need to consider the different cases depending on the sign of the expression inside the absolute value (x + 1).
- For positive values of (x + 1), the absolute value of (x + 1) is just (x + 1) itself. So, g(x) = x + 1.
- For negative values of (x + 1), the absolute value of (x + 1) is equal to the opposite of (x + 1), which gives us -(x + 1). Hence, for negative (x + 1), g(x) = -(x + 1).
Therefore, the piecewise function for g(x) = |x + 1| can be written as:
g(x) = {
x + 1, if x + 1 >= 0,
-(x + 1), if x + 1 < 0
}
Remember to simplify further if possible.