# What is the range of the function f(x)= abs(x-1) + x-1?

Jul 29, 2018

Range of $| x - 1 | + x - 1$ is $\left[0 , \infty\right)$

#### Explanation:

If $x - 1 > 0$ then $| x - 1 | = x - 1$ and $| x - 1 | + x - 1 = 2 x - 2$

and if $x - 1 < 0$ then $| x - 1 | = - x + 1$ and $| x - 1 | + x - 1 = 0$

Hence, for values $x < 1$, $| x - 1 | + x - 1 = 0$ (also for $x - 0$).

and for $x > 1$, we have $| x - 1 | + x - 1 = 2 x - 2$

and hence $| x - 1 | + x - 1$ takes values in the interval $\left[0 , \infty\right)$ and this is the range of $| x - 1 | + x - 1$

graph{|x-1|+x-1 [-10, 10, -5, 5]}