Question

calculus help

Get the conditions of the absolute value function

f(x) = |x| + |x - 1|

I can individually get the conditions of each term, using the rule |x| = x >= 0 and |-x| x < 0

Im having a hard time getting the specific condition of the whole function, help is much appreciated :)

Answer 1

By the definition of the absolute value,
\[|x|=\begin{cases}x&\text{for }x>0\\0&\text{for }x=0\\-x&\text{for }x<0\end{cases}\]
and more generally,
\[|x-c|=\begin{cases}x-c&\text{for }x>c\\0&\text{for }x=c\\-(x-c)&\text{for }x<c\end{cases}\]
Setting \(c>0\), if we add the two, we have 5 cases:
\[\color{red}{|x|}+\color{blue}{|x-c|}=
\begin{cases}\color{red}x+\color{blue}{x-c}&\text{for }x>c\\
\color{red}c+\color{blue}0&\text{for }x=c\\
\color{red}x+\color{blue}{c-x}&\text{for }0<x<c\\
\color{red}0+\color{blue}{c-x}&\text{for }x=0\\
\color{red}{-x}+\color{blue}{c-x}&\text{for }x<0\end{cases}\]
Simplifying a bit, you have
\[|x|+|x-c|=\begin{cases}2x-c&\text{for }x>c\\
c&\text{for }x=c\\
c&\text{for }0<x<c\\
-(x-c)&\text{for }x=0\\
-(2x-c)&\text{for }x<0\end{cases}\]

Answer 2

Thanks much , sorry for the late acknowledgement :)