b To write the absolute value function as a , we first need to determine where the function changes direction, as this is where our will split. In this case, this occurs when x=3.
We will let the domain of the left piece be all less than
3.
Domain Left Piece:3<x
The right piece of the function will then have a domain that includes all greater than or equal to
3.
Domain Right Piece:x≥3
Next, we can write the function as two separate equations by splitting our absolute value into its two cases: one positive and one negative.
f(x)={2[-(x−3)],2(x−3),x<3x≥3
We can simplify these equations a little bit. Let's start with the negative case.
Now we can simplify the positive case.
f(x)=2(x−3)⇔f(x)=2x−6
Finally, we can write the equation of the piecewise function.
f(x)={-2x+6,-2x−6,if x<3if x≥3