Question

Find the derivative and domain of the function: f(x)=|x+1| + |x| + |x-1|

Answer 1

As you know, |x| is just x for positive (or zero) x, and -x if x is negative.
(This "makes neg numbers positive": |-3| = --3 = 3).
For function f this means:
If x < 0, replace |x| by -x. For x >=0, just write x.
Also, for x-1<0, replace |x-1| by -x+1
Same applies for x+1.

Now what does this all lead to?
If you look carefully, you can see that the domain, which consists of all real numbers, (after all, you can take the absolute value of any real number!), is split up into 4 parts:

1. x < -1
2. -1 <= x < 0
3. 0 <= x < 1
4. x >=1

For each of these parts, there is a different formula for f, due to the replacing of |x| by x or -x and so on.

This is the formula for part 1:

f(x) = -x-1 -x -x+1 = -3x

f'(x) = -3.

Explanation: In part 1 each of the numbers between the || is negative, so each has to be replaced with -x, -(x+1) and -(x-1).

All you have to do now is to determine f(x) for the other parts. These are all simple linear functions, so differentiating them is a breeze.

Also, bear in mind that because of the sign-switching in -1, 0 en 1, f is not differentiable there!

Can you find out the other 3 formulas for f?

Answer 2

I'm still not really sure how to find the other parts...How do you figure out which segment is supposed to be negative and which is supposed to be positive?

Answer 3

Actually, I figured it out. Thank you!

Answer 4

E.g. part 2: -1 <= x < 0
There, x is negative, so replace |x| with x.
Also, x-1 is negative so replace with -(x-1)

YW!