# How do you express  y=|x^2-9|+|x^4-16|+|x^6-1|, sans the symbol |...|?

Dec 14, 2016

$y = \sqrt{{\left({x}^{2} - 9\right)}^{2}} + \sqrt{{\left({x}^{4} - 16\right)}^{2}} + \sqrt{{\left({x}^{6} - 1\right)}^{2}}$

#### Explanation:

$\left\mid x \right\mid = \sqrt{{x}^{2}}$

This is a one dimensional version of the distance formula.

So:

$y = \sqrt{{\left({x}^{2} - 9\right)}^{2}} + \sqrt{{\left({x}^{4} - 16\right)}^{2}} + \sqrt{{\left({x}^{6} - 1\right)}^{2}}$

graph{y = sqrt((x^2-9)^2)+sqrt((x^4-16)^2)+sqrt((x^6-1)^2) [-2.5, 2.5, 18.5, 30]}

If you want to cover Complex values too use:

$\left\mid z \right\mid = \sqrt{z \overline{z}}$

So:

$y = \sqrt{\left({x}^{2} - 9\right) \overline{\left({x}^{2} - 9\right)}} + \sqrt{\left({x}^{4} - 16\right) \overline{\left({x}^{4} - 16\right)}} + \sqrt{\left({x}^{6} - 1\right) \overline{\left({x}^{6} - 1\right)}}$

Dec 15, 2016

Four piecewise definitions can be given. See explanation..

#### Explanation:

graph{-x^2-x^4+x^6+24 [-80, 80, -40, 40]}

The given equation is the combined form, for the four piecewise

definitions

$y = - \left({x}^{2} - 9\right) - \left({x}^{4} - 16\right) - \left({x}^{6} - 1\right)$

$= - {x}^{2} - {x}^{4} - {x}^{6} + 26 , x \in \left[- 1 , 1\right]$

$= - \left({x}^{2} - 9\right) - \left({x}^{4} - 16\right) + \left({x}^{6} - 1\right)$

$= - {x}^{2} - {x}^{4} + {x}^{6} + 24 , x \in \left[- 2 , - 1\right] \mathmr{and} \left[1 , 2\right]$

$= - \left({x}^{2} - 9\right) + \left({x}^{4} - 16\right) + \left({x}^{6} - 1\right)$

$= - {x}^{2} + {x}^{4} + {x}^{6} - 8 , x \in \left[- 3 , - 2\right] \mathmr{and} \left[2 , 3\right]$

$= \left({x}^{2} - 9\right) + \left({x}^{4} - 16\right) + \left({x}^{6} - 1\right)$

$= {x}^{2} + {x}^{4} + {x}^{6} - 26 , x \le - 3 \mathmr{and} x \ge 3$.

As I am unable to get the all-in-one graph, the first piece can be

seen as the central part of the inserted second graph, with zenith at

(0, 26), and ends at $\left(\pm 1 , 23\right)$

The second is in the first graph, with ends at (+-24) and (+-2,

68). Likewise, interested readers can secure the other two pieces.

The common points $\left(x = \pm 1 , \pm 2 , \pm 3\right)$ of the pieces are

( 2-tangent ) nodes of this continuous function.

graph{sqrt(-x^2-x^4-x^6+26)^2 [-80, 80, -40, 40]}