Question

How do I write this without modulus sign?

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

Answer 1

`f(x) = sqrt((x-1)^2)+sqrt((x+1)^2)`

or more generally:

`f(x) = sqrt((x-1)bar((x-1)))+sqrt((x+1)bar((x+1)))`

Explanation:

For real values of `t` we have:

`abs(t) = sqrt(t^2)`

So if `x` is real valued then we can write:

`f(x) = sqrt((x-1)^2)+sqrt((x+1)^2)`

For complex values of `t` we have:

`abs(t) = sqrt(tbar(t))`

So if `x` can be complex valued, then we can write:

`f(x) = sqrt((x-1)bar((x-1)))+sqrt((x+1)bar((x+1)))`

Alternatively, we can write it as a piecewise function.

Note that in general:

`abs(t) = { (t " if " t >= 0), (-t " if " t < 0) :}`

So:

• If `x < -1` then `abs(x-1)+abs(x+1) = -(x-1)-(x+1) = -2x`

• If `-1 <= x < 1` then `abs(x-1)+abs(x+1) = -(x-1)+(x+1) = 2`

• If `1 <= x` then `abs(x-1)+abs(x+1) = (x-1)+(x+1) = 2x`

So:

`f(x) = { (color(black)(-)2x " if " x < -1), (color(white)(-)2color(white)(x)" if " -1 <= x < 1), (color(white)(-)2x " if " 1 <= x) :}`

graph{sqrt((x-1)^2)+sqrt((x+1)^2) [-5.003, 4.997, -0.56, 4.44]}