Help with an absolute value function?

Question

I have a problem for my calculus class that I don't know how to do. The instructions say:

Graph f(x) = |x| + |x + 2| using a graphing utility then rewrite the function without absolute value signs.

I've graphed the function on my calculator, but I don't know how to rewrite it. Any help would be greatly appreciated.

Answer 1

"Rewrite the function without absolute value signs" are terrible directions, but some books use this phrasing.

What they mean is "rewrite as a piecewise function, without using absolute value."

When x < -2, |x| + |x + 2| = -x - x - 2 = -2x - 2.

When -2 <= x < 0, |x| + |x + 2| = -x + x + 2 = 2.

When x >= 0, |x| + |x + 2| = x + x + 2 = 2x + 2.

So the answer they are probably looking for is

f(x) =

-2x - 2, if x < - 2

2, if -2 <= x < 0

2x + 2, if x >= 0.

Answer 2

One way to rewrite an absolute value |x| is √(x²)

Your equation would become √(x²) + √((x + 2)²)